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Application of surface-renewal-stretch model for interface mass transfer
Chemical Engineering Science 61 (2006) 3917 – 3929 www.elsevier.com/locate/ces
Application of surface-renewal-stretch model for interface mass transfer Babak Jajuee, Argyrios Margaritis ∗ , Dimitre Karamanev, Maurice A. Bergougnou Department of Chemical and Biochemical Engineering, Faculty of Engineering, University of Western Ontario, London, Ont., Canada N6A 5B9 Received 4 April 2005; received in revised form 8 December 2005; accepted 20 January 2006 Available online 9 March 2006
Abstract A new surface-renewal-stretch (SRS) model was developed to correlate experimental data for the time-average overall mass transfer coefficient, KL,av , in liquid–liquid and gas–liquid mass transfer systems. The model is based on the equation of continuity, which includes both turbulent and convective mass transfer at the liquid–liquid and gas–liquid interfaces. The model incorporates Dankwerts surface-renewal model with the penetration theory for surface stretch proposed by Angelo et al. [Angelo, J.B., Lightfoot, E.N., Howard, D.W., 1996. Generalization of the penetration theory for surface stretch: application to forming and oscillation drops. A.I.Ch.E. Journal 12 (4) 751–760]. We used our new SRS mass transfer model to correlate successfully the existing interface mass transfer experimental data from published literature. As a result, the experimental mass transfer coefficient data was predicted with a high degree of accuracy. 䉷 2006 Elsevier Ltd. All rights reserved. Keywords: Mass transfer coefficient; Interface; Bubble column; Gaslift contactor; Airflit bioreactor; Absorption; Bubble; Drop
1. Introduction Mass transfer theories are mainly developed for the process of absorption of a gas into a liquid, even though their application might be extended to the cases that mass transfer occurs between any two immiscible fluid phases. Before turning to the principles involved, the reader should be aware of certain terminology which is basic to understanding the material presented in this paper. For our purposes, a “fluid particle” is a self-contained body with maximum dimension between about 0.5 m and 10 cm, separated from the surrounding medium by a recognizable interface. The material forming the particle and the medium surrounding the particles will be termed the “dispersed phase” and the “bulk phase”, respectively. If the dispersed phase is in the liquid state, the particle is called a “drop”, and if it is a gas, the particle is referred to as a bubble. Together, drops and bubbles comprise “fluid particles”. Henceforth, we use “continuous phase” to refer to the “bulk phase” with convective motion. Moreover, consideration is limited in the first place to the cases in which the bulk phase and the ∗ Corresponding author. Tel.: +1 519 661 2146; fax: +1 519 661 4275.
E-mail address: [email protected] (A. Margaritis) URL: http://www.eng.uwo.ca/people/amargaritis/. 0009-2509/$ - see front matter 䉷 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2006.01.026
dispersed phase are both a Newtonian fluid and then it is extended to cases in which the bulk phase would be also non-Newtonian. The conventional and simplest picture of mass transfer between two fluid phases, borrowed from a similar concept used in the convective heat transfer, is that there exists a stagnant film at the interface (Lewis and Whitman, 1924). In this case, the surface concentration may be assumed to be the saturated ∗ , while the concentration of the bulk phase is concentration, cA kept uniform outside this film by turbulent mixing. The rate of mass transfer per unit area, also known as mass transfer flux, will then be constant (steady state) and given by the following expression: NA =
D (cA,i − cA0 ), zF
(1)
where NA is the net flux defined as the rate of absorption per unit area, cA,i is the concentration at the interface, cA0 is the bulk concentration, D is the molecular diffusivity and zF is the effective film thickness which depends upon the nature of the flow conditions. Therefore, the net flux appears to conform to the expression NA = kL (cA,i − cA0 ),
(2)
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where kL is the film mass transfer coefficient and is constant for a fluid–fluid system under given conditions. Although it is preferable to refer to the conventional picture due to its simplicity as though it really exists, the fictitious nature of the film theory may lead to erroneous results in many actual mechanisms of mass transfer. In particular, when fluid particles rise or fall in infinite media with turbulent motion, the conditions required maintaining a stagnant film at the surface of moving fluid particles is lacking. In other words, it appears unlikely that the surface layer on each particle keeps its identity throughout all times. Higbie (1935) took a major step forward by introducing the “penetration theory”. This theory emphasizes that in many situations the time of exposure of a fluid to mass transfer is too short to let the concentration gradient of the film, characteristic of steady state, to develop. With the assumption of constant but short time of exposure for all fluid particles, they are subject to unsteady state diffusion or penetration of solute. Taking the uniform initial concentration of cA0 for the dissolved component and the surface concentration of cA,i , as soon as the bulk phase is exposed to the diffusing component, which may be also taken as the equilibrium solubility of it in the bulk of the fluid, the initial and boundary conditions become cA0 at = 0 for all z, (3) c = cA0 for all z = ∞, cA,i for all z = 0, in which is the time of exposure to the diffusing component. Thus, the average flux over the time of exposure is 4D NA,av () = (cA,i − cA0 ). (4) This theory pictures an unbroken film for component diffusion while the bulk phase is stagnant and has constant time of exposure for all eddies and fluid particles; nevertheless, the application of the penetration theory is probably extended to cases where the bulk phase is also in turbulent motion. It can be seen that the rate of absorption becomes very slow after a time unless the surface is renewed by stirring or by convection. This was first clarified by Dankwerts (1951) with “surface-renewal theory” as he pointed out that Higbie theory with its constant time of exposure of eddies at the surface is a special case of what may be a more realistic picture of the processes occurring during absorption into an agitated fluid where in fact eddies are exposed for varying lengths of time. In the surface-renewal theory, the interface is pictured then as a mosaic of different exposure-time surface elements where eddies are continually exposing fresh surfaces to the diffusing component and sweeping away and mixing into the bulk of the fluid. Dankwerts assumed that any portion of the bulk phase absorbs the component at a rate given by Eq. (4) in a case that the depth of penetration of the solute would be much smaller than the scale of turbulence, so that the intensity of turbulence accentuates velocity gradient existing beneath the surface and at the same time decreases the depth of penetration by shortening the period for any part of the bulk phase which is exposed to the diffusing
component before being submerged once again. To compensate for the lack of turbulence at the interface, Dankwerts added a new constant parameter called the mean rate of production of fresh surface, s, to the penetration theory and assumed that the chance of a surface element being replaced by another within a given time is independent of how long it has been in the surface; hence the fractional rate of replacement of the elements belonging to different age groups is equal to s. Combination of the continuity equation with the same conditions of Eq. (3) and the concept of surface renewal gives a new expression to the mean rate of mass transfer per unit area of turbulent surface √ (5) NA,av = Ds(cA,i − cA0 ). The penetration and surface-renewal theories tacitly assume that immediately underlying all freshly formed surfaces is fluid with the same concentration, cA0 , of dissolved component. It is equal to consider the surface elements to be infinitely deep that diffusing component never reaches the region of constant concentration below. Dobbins (1956) pointed out that this is not true for aeration systems where flowing streams continually absorb oxygen and the concentration of the bulk of the fluid changes. Accordingly, Dobbins considered a finite depth of surface elements and eddies by replacing the third boundary condition in Eq. (3) with cA = cA0 for z = za and obtained ⎡ ⎤ 2 √ sz a⎦ (6) (cA,i − cA0 ). NA,av = ⎣ Ds coth D n , with n It can be seen that the general dependence, KL ∝ DAB dependent upon circumstances, is satisfied in this expression. Other researchers have made similar suggestions (King, 1966; Kozinski and King, 1966; Toor and Marchello, 1958; Lamont and Scott, 1970). In summary, for rapid penetration (D), or for thin surface elements, and small rate of surface renewal (s), the situation is close to steady state and the predominant character of the mass transfer is described by the film theory; whereas for slow penetration, or for short time of exposure or rapid renewal (unsteady state situation) it follows Eqs. (5) or (6). Remarkably, Eq. (5) may be applied for bulk phase moving parallel to the surface with a velocity that varies with the depth while the time of exposure is so short that the depth of penetration is much smaller than the depth at which the velocity is appreciably different from that at the surface. As a complimentary to this, Eq. (6) depicts a more realistic picture of the penetration depth by assuming a finite depth of surface element for fluid particles. Molecular diffusivity is the most important attribute in concentration changes among unsteady state theories discussed above. In addition to transport by molecular motion, however, mass may also be transported by the bulk motion. Therefore, a comprehensive net flux of mass transfer in the liquid including bulk motion is the combined net flux vector, which consists of the both molecular flux vector and convective flux vector
N A = JA + cA v,
(7)
B. Jajuee et al. / Chemical Engineering Science 61 (2006) 3917 – 3929
where JA and v are the molar molecular flux vector and the molar average velocity vector, respectively. A, JA = −cD ∇x v =
N
xA vA ,
(8)
A=1
where ∇ is the vector differential operator and is defined in rectangular coordinates as ∇ = x
j j j + y + z jx jy jz
(9)
in which is the unit vector. Angelo and coworkers (Angelo et al., 1966; Angelo and Lightfoot, 1968; Stewart et al., 1970) developed the penetration concepts in the absence of turbulence when there is bulk motion by introducing a promising theory called “surface-stretch theory”. On the assumption that diffusion happens in the same direction as convection, i.e., z, the surface-stretch theory expresses the local velocity vz in time-dependent interface surface A, and gives the continuity equation a new form −z
j ln A j
jcA jcA j 2 cA + =D 2 j jz x,y jz
(10)
whose solution leads to another expression for the net flux ⎤ ⎡ ⎢ (A(/0 )) D/0 ⎥ NA,av = ⎣ (11) ⎦ (cA,i − cA0 ), /0 2 A () d 0 where A is the interfacial surface through which mass transfer takes place while it changes periodically with time, and 0 is a constant with the dimension of time defined for each system. Eq. (10) will be justified in the discussion of surface-renewalstretch theory. In brief, the concepts of bulk motion and turbulent motion have been neglected in the surface-renewal and surface-stretch theories, respectively. Presented here is a holistic approach to the treatment of mass transfer processes occurring in fluid–fluid contactors, while the symbiotic relationship exist between turbulent and bulk motions synchronously. 2. Fluid–fluid contactors More recently, fluid–fluid contactors are referred to a myriad of configurations developed for two or three-phase applications in which fluid particles, e.g. bubbles or drops, have interactions. Fluid particles oscillate and wobble when rising through a denser media and make surfaces wavy and rippled. Gaslift systems considered as gas–liquid contactors, for example, are widely used in many operations such as fermentation and wastewater treatment especially where microorganisms are used as biocatalysts. In general, they are pictured as pneumatically agitated devices characterized by circulation in a defined cyclic pattern where slurry or substrate circulates from the riser
3919
to the downcomer section. Aeration of one section exploits hydrostatic pressure differences by making lower hydrostatic head compared to the non-aerated section at the same vertical coordinate across the membrane wall (King, 1966). Unsteady circulation with rocking motion of fluid particles increases the intensity of turbulence, which in turn changes incessantly the conformation, and position of eddies and exposes fresh surfaces to fluid particles. At the same time, bulk motion also drives eddies and liquid particles in a continuous circulation. It is well known that the mass transfer processes occurring in fluid–fluid contactors are too complicated to be explained by an implausible theory. On the other hand, providing a thorough picture of mass transfer, for many purposes, is impossible without taking the inevitable existence of both bulk and turbulent motions into account. 3. Theoretical development 3.1. Surface-renewal-stretch (SRS) model The main objective of this work is to extend the penetration theory of mass transfer to fluid–fluid interacting systems, e.g. gaslift contactors, in which both turbulent and bulk motions exist. Consider a bulk phase with uniform rate of mass transfer through the interface that is maintained in turbulent and bulk motion, as shown in Fig. 1. This case may be expounded from both macroscopic and microscopic standpoints. The former is pictured with eddies and fluid particles that have an explicit bulk motion, while the latter with the surface of each element that is continuously replaced with fresh surface and has a distribution of ages varying from zero to infinity at any particular instance. Let the total area of the surface exposed to the gas would be equal to A∞ . On the assumption that the chance of any portion of surface element being replaced by another is quite independent of how long it has been in the surface, the average rate of exchange of the film which produces fresh surfaces is taken to be constant and equal to r(). The concept of surface–age distribution function might be borrowed from the surface-renewal theory, in the sense that at any moment the distribution of ages of the film area is given by the function (), and the film area comprising the elements having ages between and + d is () d. This does not vary with time at steady state and is equal to the film area in the age group between − d and less the portion of this, which is replaced by fresh surface in a short time interval equal to d
r () d = ( − d) d 1 − d . (12) A∞ Writing Taylor series for ( − d) gives () = () −
r d() () d. d − d A∞
(13)
Thus r d() =− () d A∞
(14)
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B. Jajuee et al. / Chemical Engineering Science 61 (2006) 3917 – 3929 vz
Distance, z
Finite depth of element surface
cA0 For all
za At z=0 for all > 0 cA,i
Concentration
Fluid Particle
Fig. 1. Mass transfer mechanism in a fluid–fluid contactor when the scale of turbulence and convection are both taken into account.
and since ∞ () d = A∞
(15)
0
the distribution function can be obtained as () = re−s ,
(16)
where s has the value of r/A∞ , defined as the rate of production of fresh surface which is exposed to the gas (Dankwerts, 1951). The area of the mass transfer surface, () d, is a function of time and may be readily integrated to yield A() = −A∞ e−s + C1 .
(17)
Taking A = A0 at = 0 gives A() = A∞ (1 − e−s ) + A0 .
(18)
This expression represents the area of the mass transfer surface changes with time and the intensity of turbulence, s. On the other hand, this area may also change with time due to convection, which produces turbulent motion in turn. We limit consideration to situations in which diffusion happens in the direction of convection, perpendicular to the interfacial surface area. We shall also assume that at any given instant an interface, which expands and shrinks with time, exists between the continuous fluid and fluid particles rising in the z direction with the bulk motion, as shown in Fig. 1. Under these circumstances, diffusion is negligible in directions parallel to the interface and is also considered one dimensional in the neighborhood of any surface element. The continuity equation, therefore, takes the form vz
jcA jcA j 2 cA + =D 2 , jz j jz
(19)
where vz may be derived by making an overall momentum balance over each contactor system; however, it is also attributable to the corresponding interfacial area of the time-dependent surface. In the case of constant molar density of the solution, consider a small flat element of surface xy moving with the fluid bulk velocity. The direction of the z-axis is pointing into the diffusing phase whose net rate of mass transfer is assumed to be small. The area of the surface of this element changes with time
jA = [x(vy |y+y − vy |y ) + y(vx |x+x − vx |x )]. j ave (20) For any point in the interfacial surface, we may divide Eq. (20) by xy and take the limit as x, y approach zero
jvy j ln A jvx = + . (21) j jx jy When the density is constant this expression may be written with the aid of the equation of continuity as
j ln A jvz =− . (22) j jz Since the expression on the left-hand side is independent of z, we may write for negligible net transfer across the interface
j ln A vz = −z , (23) j where the overall interfacial area A() is obtainable from Eq. (18) at any time. It is worth noting that z should be small compared to the local radii of curvature for curved surfaces like the interfacial surface of a gas bubble. This is consistent with the limitations on the penetration theory. Eqs. (19) and (23) are
B. Jajuee et al. / Chemical Engineering Science 61 (2006) 3917 – 3929
then combined to give Eq. (10) −z
j ln A j
jcA jcA j 2 cA + =D 2 . j jz x,y jz
(10)
I.C. c = cA0
at = 0 for all z,
(24)
B.C.1
c = cA0
for all at z = za ,
(25)
B.C.2
c = cA,i
for all at z = 0.
(26)
The initial condition and the first boundary condition above follow the assumed uniform bulk concentration of gas in entire solution and finite depth of the surface elements or eddies. Nevertheless, the rough assumption of infinitely deep surface element is constantly referred to as though there are short constant times. This may be regarded as a harmless and convenient usage for many systems but not for aerated flowing systems in which bulk concentration continually changes with time. In airlift systems, the dissolving solute is able to reach the depth za corresponding to the thickness of the eddy, so that from the solute point of view, za is essentially finite. The second boundary condition expresses the concentration in the continuous phase at the interface, cA,i , which may be taken as the equilibrium solubility of the fluid particles in the fluid. The concept of each concentration has been illustrated schematically in Fig 1. To solve Eq. (10) we may define the following dimensionless variables: C=
cA − cA,i , cA0 − cA,i
Eq. (30) is a partial differential equation whose easiest way of solving is considering a combination of variables in order to convert it to a simple differential equation C = C(),
For simplicity sake, we shall assume that D would be constant. The boundary conditions over Eq. (10) then are
(27)
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= Z × G(T ).
Hence, Eq. (30) becomes F G dC d2 C + − 3 = 0, G2 G d d2
(35)
(36)
where G indicates the derivative of G with respect to T at constant Z. The principle of “combination of variables” requires G F = , (37) − G2 G3 where is an arbitrary constant. Eq. (36) is thus simplified to d2 C dC + = 0. 2 d d
(38)
In brief, Eq. (30) was reduced to Eq. (38), which is an ordinary differential equation and has the solution of √ √ 2 C() = C2 + C3 × erf . (39) 2 If the assumed combination of variables is successful it should be necessary that Eq. (39) satisfy all Eqs. (31)–(33). Application of the initial condition and the two boundary conditions permits the evaluation of C2 and C3 which are constant C(0) = 0,
(40)
C(A ) = 1,
(41)
z Z= √ , D0
(28)
where A is the value of when Z = ZA . Eq. (37) may be rearranged more simply as
, 0
(29)
F G − G = G3 .
T =
where 0 is a constant with units of time and is evaluated with time characteristic of each system; e.g. for a gaslift contactor 0 might be bubble-formation time. The dimensionless form of our system becomes jC jC j2 C , − F (T )Z = jT jZ jZ 2
(30)
C = 1 at T = 0 for all Z,
(31)
C=1 C=0
for all T at Z = ZA = √ for all T at Z = 0,
za , D0
F (T ) =
j ln A(T ) jT
C() =
erf() . erf(A )
(43)
Eq. (42) is a Bernoulli equation and, therefore, by the substitution G = h−1/2
(44)
and the integrating factor (33)
e2
F (T ) dT
= e2 (j ln A/jT ) dT = A(T )2
(45)
it takes the form
. x,y
If the chosen combination of variables is to be successful, Eqs. (40) and (41) represent restrictions that should be met. For convenience we take = 2, thereby giving
(32)
where
(42)
(34)
A2
dh dA = −2hA + 4A2 . dT dT
(46)
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B. Jajuee et al. / Chemical Engineering Science 61 (2006) 3917 – 3929
We may readily solve Eq. (46) whereby it follows that A(T ) , G(T ) = T 4 0 A2 () d + C4
(47)
where C4 is a constant that must be zero to satisfy Eq. (41). Thus, A(T ) . G(T ) = T 2 4 0 A () d
(48)
To obtain the complete solution of Eq. (30) we should recall that = Z × G(T )
T 2 erf Z × A(T ) 4 0 A () d
, C= (49) T 2 erf ZA × A(T ) 4 0 A () d where ZA is a constant regarding to za in which C =1. This expression satisfies Eqs. (32)–(33). It also automatically satisfies Eq. (31) if one substitutes A(T ) from Eq. (18) into Eq. (49). Returning to the concept of age–surface distribution function, the instantaneous rate of absorption for surface elements having age and film area () d is (T ) dT , dNA,z = jA,z ∞ 0 (T ) dT where jA,z = −D hence
(50)
jZ jcA jC = (c − c )D A,i A0 jz z=0 jZ Z=0 jz
(51)
Ds 2 0 dNA,z = (cA,i − cA0 ) ⎡ ⎤ T 2 () d A(T ) A 0 ⎢ ⎥
⎥ ×⎢ ⎣ ⎦ T erf ZA A(T ) 4 0 A2 () d × exp((−s0 )T ) dT = (cA,i − cA0 )KL (T ) dT ,
(52)
where KL is the local mass transfer coefficient at a given time on the interface. Hence the average flux over the time of exposure per unit area of turbulent surface is ∞ NA,av = (cA,i − cA0 ) KL (T ) dT 0
= KL,av (cA,i − cA0 ).
(53)
where n is a constant equal to the ratio of A0 /A∞ which is essentially less than unity and greater than zero, and is a dimensionless variable taken as equal to s0 T . The price that is paid for a greater accuracy in analytical models is an increase in complexity. The above analysis represents a comprehensive expression for the average mass transfer coefficient in any fluid–fluid system in which turbulent motion, bulk motion, and a finite depth of penetration exist. All previous mass transfer theories discussed earlier can be developed as special cases of the above treatment. It can be seen that the overall mass transfer coefficient is a function of the diffusion coefficient, D, the fractional rate of surface renewal, s, and the depth of penetration, za . The main drawback to such a complicated expression for mass transfer coefficient is having an error function in denominator which makes the integral of Eq. (54) too convoluted to be employed for practical purposes; nevertheless, in many cases some assumptions help to circumvent the difficulty of the problem. When the scale of turbulence is high, as in the case of many practical systems, the rate of production of fresh surface increases and produces infinitely deep liquid elements corresponding to the thickness of eddies due to the short exposure times. Hence za approaches infinity and the error function becomes close to unity. Moreover, regarding the fact that 0 < n < 1, the integrand in Eq. (54) finds virtually the value of 2 (Maple, 2003) when the error function is equal to unity, and thus 4Ds KL,av = . (55) This is a new model of time-average overall mass transfer coefficient coming out of the equation of continuity in the case that the two immiscible fluid phases undergo turbulent motion with one-dimensional diffusion and convective motion in the same direction. It is noteworthy to recall that Eq. (49) was achieved by ascribing change of time-dependent exposed surface area and increase of intensity of turbulence to convection motion, as expressed in Eq. (23), where vz is convective molar average velocity. As a result, s is a representative of the scale of turbulence vis-à-vis convective motion that renews the surface of the liquid by either stirring or convection, and hence Eq. (55) embodies the influence of both turbulent and convective motions. In a special case that mass transfer takes place into a stagnant fluid of infinite depth, Eq. (55) changes to the model of penetration theory. The time-average overall mass transfer flux becomes 4Ds NA,av = (cA,i − cA0 ). (56) Provided no chemical reaction occurs, a different significance is merely attached to the overall mass transfer coefficient while all expressions derived for the rate of mass transfer are similar to that based on the film theory. For systems of practical
To solve Eq. (53) through Eq. (52), A(T ) should be substituted from Eq. (18) into Eq. (52) and integrated when T changes from zero to infinity. − + (1 + n))e− ∞ (−e −0.5e−2 + 2(1 + n)e− + (1 + n)2 − 1.5 − 2n Ds d, KL,av = 0 erf (s × z2 )/4D(−e− + (1 + n)) −0.5e−2 + 2(1 + n)e− + (1 + n)2 − 1.5 − 2n a
(54)
B. Jajuee et al. / Chemical Engineering Science 61 (2006) 3917 – 3929
3923
importance, average mass transfer coefficients can be evaluated experimentally in such an easy way without being concerned about parameters like D, s, za , if the average flux is obtained from the general expression of Eq. (53). Having selected an appropriate system, the equation of continuity for species A in a binary system is written over the volume of a shell in the form
than 0.5, where H is the average distance to where eddies approach the interface. In this case, the length and velocity scales of the eddies for isotropic turbulence can be written as 3 1/4 l= , v = ( ˙)1/4 , (61) ˙
dNA,z dc + (1 − p ) = 0. dz d
where = / is kinematic viscosity and ˙ is viscous dissipation of energy per unit mass of eddies. The connotation of Kolmogoroffs’ time scale of eddies in viscous dissipation range can be used in the jargon of surface-renewal theory, in the sense that s is defined (Banerjee et al., 1968)
(57)
It should be noted that Eq. (57) is only valid when absorption takes place in the z direction with no chemical reaction. Assuming jz ≈ z = a −1 , it may be written at any given instant as −KL a(cA,i
dc − c) + (1 − p ) = 0, d
(58)
where a is the specific surface area, i.e., the interfacial surface area divided by the bulk volume of fluid–fluid contactors. The negative sign of the expression on the left-hand side indicates that the net rate of mass transfer changes inversely with depth. We may readily solve Eq. (58) by rearranging and getting an integral from zero to time c KL a 0 d dc = (59) (1 − p ) c0 (cA,i − c) then, ln
cA,i − c (−KL a) = . cA,i − cA0 (1 − p )
(60)
The experimental data might be obtained to evaluate KL a from Eq. (60), while finding a good correlation for s as a function of measurable parameters, e.g. fluid–fluid skip velocity, superficial velocities of two fluid phases, and diameter of fluid particles, provides a good picture based on the surface renewalstretch model to predict the behavior of fluid–fluid contactors in different situations. On the evidence of literature, Eq. (60) seems virtually ambiguous by being directly derived from the film theory. This happens because the net flux of mass transfer in the film theory also follows the same general expression as in Eq. (53). The accuracy of Eq. (55) can be easily scrutinized by analyzing the effects of the scales of turbulence and convection motion on the overall mass transfer coefficient. 3.2. Mass transfer coefficient correlation for gas bubbles Consideration is limited in the first place to the cases in which the dispersed phase is in the gas state. Due to its great acceptance, the determination of the mean fractional rate of surface renewal, s, through physical and hydrodynamic arguments has been the subject of much research interest (Banerjee et al., 1968; Skelland and Lee, 1981). In this work, the value of the mean rate of production of fresh surface has been determined without recourse to observations of mass transfer rates, as used in previous studies (Banerjee et al., 1968). Harriott (1962) showed that the concept of surface-renewal the√ ory could only be expected when the group H / D/s is less
s≡
v . l
Substitution of Eq. (61) for v and l into Eq. (62) yields ˙ s= .
(62)
(63)
Energy dissipation rate is calculated as follows (Wang et al., 2005):
g ˙ = Usg − (64a) Usl g. 1 − g In bubble columns and gaslift systems with low superficial liquid velocity (Usl ≈ 0) the energy dissipation rate can be approximated as ˙ = Usg g.
(64b)
Combination of Eqs. (63) and (55) provides an expression for s, and hence the time-average overall mass transfer coefficient becomes 1/2 4D Usg g . (65) KL,av = In deriving this expression it has been assumed that the contacting phases are Newtonian fluids with gas bubbles composing the dispersed phase and that the rate of surface renewal and the rate of energy dissipation follow the Eqs. (62) and (64b), respectively. The latter is almost certainly an oversimplification, but the approximations are probably no more drastic than that involved in real cases of mass transfer. Reviewing the literature gives a general indication of the behavior which is expected from this type of systems. Representing the non-Newtonian behavior of fluids by the power-law model, = K n leads to the expression 1/2
4D Usg g 1/(1+n) . KL,av = Similarly the analogs of Eq. (61) are 3 1/2(1+n) , v = ( ˙n )1/2(1+n) l = 2−n ˙
(66)
(67)
(68)
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B. Jajuee et al. / Chemical Engineering Science 61 (2006) 3917 – 3929
Table 1 A summary of investigator designs and operational variables for Fig. 2 Reference
Liquid
Dc (m)
× 103 0.92
991 995 1030 1130 1230
(Pa s)
(kg/m3 )
× 102
C8
Ave. Abs. Dev. from Eq. (80) (%)
7.02
1.326 1.571
15.2 28.04
7.1 7.1 7.1 7.1
0.851 1.044 8.13 0.512
21.19 1.65 26.1 50.44
7.26
0.52
22.83
(N/m)
Kawase et al. (1987)
Water
0.23 0.76
Nakanoh and Yoshida (1980)
Water 10% sucrose 30% sucrose 50% sucrose
0.1455
0.80 1.0815 2.543 10.85
Shamlou et al. (1995)
Fermentation Broth
0.317
0.894
El. Temtamy et al. (1984)
Water
0.15
1
1000
7.26
0.687
23.1
Koide et al. (1983)
Water 100 mol/m3 BaCl2 50% glycerol
0.14
0.8941 0.9185 5.93
997 1016 1141
7.196 7.242 6.688
0.682 0.846 1.496
57.1 23.73 30.09
Nakao et al. (1983)
Water
0.6
1.73
998.5
7.39
1.838
37.09
Jackson and Shen (1978)
Water
1.83 7.62
1.224 1.224
999.4 999.4
7.45 7.45
1.16 0.959
12.1 7.2
in which = K/ . Moreover, in both Newtonian and nonNewtonian solutions the average shear rate varies about proportional to superficial gas velocity (Nishikawa et al., 1977) = 5000Usg .
(69)
This has been regarded, for many purposes, as a useful relationship for derivation of mass transfer correlations (Nakanoh and Yoshida, 1980; Deckwer et al., 1982, Godbole et al., 1984). If unit volume of a two-immiscible-fluid mixture contains a dispersed phase volume of p made up of n fluid particles of diameter dp , then n = p /(dp3 /6). If the interfacial area in the unit volume is a, then n = a/dp2 . Equating the two expressions for n provides the specific area a=
6p dp
(70)
Similar correlations have been proposed by other researches in which the range of the exponent varies from 0.4 to 1 (Durst et al., 1979). Hence,
˙0.4 0.6 1− d 0.25 . a = C7 0.6 p
c
(74)
In the case of gas bubbles, Godbole et al. (1984) proposed the following correlation for gas hold up in non-Newtonian solutions: 0.6 −0.19
eff , g = 0.255Usg
(75)
where eff is defined as follows:
eff = K n−1 .
(76)
Substituting Eqs. (69) and (76) into Eq. (75) yields
and
(0.79−0.19n) . g = (0.255 × 50000.19(1−n) )K −0.19 Usg
6g a= dBM
(77)
(71)
when gas bubbles comprise the dispersed phase. For noncoalescing conditions, the size of gas bubbles produced in gas–liquid dispersion has been determined from a balance between surface tension forces and those due turbulent fluctuations (Calderbank, 1958) dBM = C5
997.1
0.6 ˙0.4 0.6
.
(72)
As a result Calderbank (1958) proposed the following expression for fluid particle size, dF M :
0.6 0.25 dF M = C6 0.4 0.6 p d . (73)
c ˙
This correlation agrees reasonably well with a vast body of literature data dealing with bubbles (Kawase et al., 1987). Assuming = 21 in Eq. (74) and combining Eqs. (64), (74), and (77) gives a new expression for specific gas–liquid interfacial area (8−n)/10
a = (0.5 × 50001−n/10 )C7
g 0.4 0.6 Usg
0.6 K 0.1
d
c
0.25 .
(78)
Combining Eqs. (78) and (67) and rearranging into dimensionless groups lead to a correlation for the volumetric mass transfer coefficient C8 0.5 (4n+17.5)/10(n+2) Sh = 5000(1−n)/10 √ Sc Re 0.6 (5n−2)/15(n+1) × Bo F r ,
(79)
B. Jajuee et al. / Chemical Engineering Science 61 (2006) 3917 – 3929
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Table 2 A summary of investigator designs and operational variables for Fig. 3 Reference
Liquid
n
Dc (m)
K (Pa s)
Kawase et al. (1987)
Carbopol-1 CMC-2
0.82 0.476
0.23
2.78 × 10−2 2.32
Nakanoh and Yoshida (1980)
0.3% CMC 0.5% CMC 1.0% CMC
0.97 0.985 0.835
0.1455
Godbole et al. (1984)
CMC-1 CMC-2 CMC-3 CMC-4 CMC-5
0.697 0.654 0.668 0.607 0.492
0.305
El. Temtamy et al. (1984)
Yeast-B 0.35% CMC
0.82 0.59
0.15 0.305
Joseph et al. (1984)
3% CMC
0.5242
× 102
(kg/m3 )
C8
Ave. Abs. Dev. from Eq. (79) (%)
(N/m)
992 992
6.92 7.11
0.522 0.109
3.16 17.37
1.15 1.626 12.11
1000 1000 1001
7.1 7.1 6.84
0.679 0.696 0.356
37.31 40.31 59.46
0.095 0.184 0.256 0.526 2.816
1000 1000 1002 1003 1005
7.3 7.25 7.32 6.84 6.77
0.492 0.349 0.262 0.192 0.086
27.33 20.65 3.41 3.47 18.19
1006 1002
9.67 7.10
0.624 0.189
12.37 4.62
1004
7.12
0.127
3.07
9 × 10−3 0.947 1.434
108
Shamlou et al. (1995) Kawase et al. (1987), Dc=0.76m Kawase et al. (1987), Dc=0.23m
5%
-2
El. Temtamy et al. (1984) Koide et al., (1983) Nakao et al. (1983) Nakanoh & Yoshida (1980) Jackson & Shen (1978), Dc=7.62m Jackson & Shen (1978), Dc=1.83m
7
Shexp
10
5%
+2
106
105 5x104 5x104
105
106
Shcalc
107
108
Fig. 2. Experimental Sherwood number for Newtonian fluids vs. predicted Sherwood number from Eq. (80).
where Sh = Bo =
KL aD 2c D gD 2c
,
Sc =
K/ Dc1−n , DU 1−n sg 2 Usg
and F r =
Dc g
Re =
2−n Dcn Usg
K/
,
.
For Newtonian fluids where n = 1, Eq. (79) reduces to C Sh = √8 Sc0.5 Re0.72 Bo0.6 Fr0.1 ,
(80)
in which C8 is the value of C8 when n in Eq. (79) is equal to unity. The exponential dependency of KL a on Usg is 0.95. It can be seen that the volumetric mass transfer coefficient is a
function of a number of variables including the superficial gas velocity, the dispersed and continuous phase properties, and the bubble size distribution. To keep our scope sufficiently broad, a comprehensive review of literature abounded with mass transfer data for both Newtonian and non-Newtonian fluids has been presented in Tables 1 and 2, respectively. The values of C8 have been determined for each set of mass transfer data through lack of reliable data for specific gas–liquid interfacial area. Accordingly, the constant C8 can be iteratively approximated as C8 = 1.022n3.3 .
(81)
If the value of the integrand in Eq. (54) were to be exactly equal to 2 and no assumption was made to obtain Eq. (80),
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B. Jajuee et al. / Chemical Engineering Science 61 (2006) 3917 – 3929
107 Kawase et al. (1987) Nakanoh & Yoshida (1980) Godbole et al. (1984) El. Temtamy et al. (1984) Joseph et al. (1984)
9% -1 9% +1
Shexp
106
105
104 104
105
106
107
Shcalc Fig. 3. Experimental Sherwood number for non-Newtonian fluids vs. predicted Sherwood number from Eq. (79).
i.e., the one in Eq. (64b), then the coefficient on the righthand side of Eq. (81) would be expected to be 1 rather than 1.022. The proposed correlation, Eq. (80), and the experimental results obtained by various workers for Newtonian fluids are compared in Fig. 2. The upper and lower dashed lines represent the negative and positive deviations from Eq. (80). The results are in satisfactory agreement over the wide range of 0.14mDc 7.62m with the average absolute deviation of only 25%. Absolute deviation is defined as |(Calculated value − Experimental value)/Experimental value| throughout this study. A very similar expression was introduced by Kawase et al. (1987) 0.8 Sh = √ Sc0.5 Re0.75 Bo0.6 Fr0.12 . (82) Considering Higbie’s penetration theory as a particular form of Eq. (55) and the theoretical basis for derivation of Eq. (82) is deemed to be the main reason for this close resemblance. Eq. (82) and other similar correlations (Nakanoh and Yoshida, 1980; Akita and Yoshida, 1973) approve the plausible picture of the model proposed in this work. Fig. 3 is also a plot of the predicted values of Sherwood number for non-Newtonian vs. those observed experimentally. Even though the exponential dependency of KL a on Usg , Dc , D, K/ , and is quite more complicated for non-Newtonian fluids, it provides much lower average absolute deviation of about 19% compared to previous published correlations (Kawase et al., 1987). Figs. 2 and 3 show how the points are evenly distributed around the diagonal.
for which the surface renewal rate would be determined in terms of hydrodynamic parameters, is sparse. Skelland and Lee (1981) suggested a periodically varying rate of surface renewal, reasoning that the condition required to maintain equal degree of turbulence at various locations in vessels for circulating drops of the dispersed phase appears to be lacking. In other words, the degree of turbulence changes and finds its maximum value near the impeller, minimum in the corners, and of intermediate strength in regions between these locations. The proposed relationship for the varying rate of surface renewal is s = sav + ϑ sin
2 , c
(83)
where c is the average circulation time of the droplets around the vessel. This concept is not obviously in harmony with Dankwerts (1951) theory with its constant surface-renewal rate and the one developed in this work. Substituting Eq. (83) into Eq. (14) yields
ϑc 2 () = C9 exp −sav + , (84) cos 2 c Table 3 The specifications of five systems studied for drops System
Dispersed phase
Continuous phase
Solute
1 2
Dimethyl siloxane Dimethyl siloxane
Heptanoic acid Heptanoic acid
3.3. Mass transfer coefficient correlation for drops
3
Dimethyl siloxane
To date, we only considered the cases in which the dispersed phase is a gas. The information on mass transfer between drops,
4 5
Ethyl acetate Benzaldehyde
Distilled Distilled Colonial Distilled Colonial Distilled Distilled
water water +20% pure cane sugar water +30% pure cane sugar water water
Heptanoic acid Heptanoic acid Heptanoic acid
B. Jajuee et al. / Chemical Engineering Science 61 (2006) 3917 – 3929
3927
10 0% -2
System 1. System 2.
0% +2
System 3. System 4.
KL,av /(ND) 0.5Exp
System 5.
1
1
10
KL,av /(ND)0.5Calc √ √ Fig. 4. Experimental KL,av / N D vs. predicted KL,av / N D from Eq. (88). Table 4 A summary of investigator designs and operational variables for Fig. 4 System
1 2 3 4 5
c × 103
d × 103
(Pa s) 0.0010 0.0018 0.0029 0.0010 0.0010
c
d
(Pa s)
(kg/m3 )
(kg/m3 )
0.0019 0.0019 0.0019 0.00046 0.0014
1000 1087 1131 1000 1000
873 873 873 894 1041
× 102
C10
(N/m)
D × 1010 (m2 /s)
Ave. Abs. Dev. from Eq. (88) (%)
0.039 0.032 0.033 0.006 0.015
6.01 5.66 4.02 6.01 6.01
4.857 × 10−7 3.190 × 10−7 4.399 × 10−7 7.037 × 10−7 7.715 × 10−7
16.72 43.24 12.50 35.95 40.14
Internal diameters of vessels: 0.21 m and 0.246 m. Diameters of impellers: 0.076 m and 0.106 m.
where C9 is obtainable from Eq. (15). Thus,
∞ A∞ ϑc 2 = exp −sav + d. cos C9 2 c 0
Taking s ≈ sav and substituting directly Eq. (86) into Eq. (55) yields (85)
This integral has been calculated and employed to evaluate the mass transfer coefficient for agitated vessels without bulk motion (Lee, 1978). In the case of agitation accompanied with bulk motion, A() should be evaluated afresh from Eqs. (84), substituted in Eq. (52), and finally integrated from zero to infinity to evaluate the time-average overall mass transfer coefficient; as presented in the case of constant surface-renewal rate in this paper. For the case that H = DT , a general correlation for sav obtained from statistical analysis method has been developed by Skelland and Lee (1981) as follows:
√ Di 0.592 1.328 0.5 sav = C10 −0.502 ReN N , (86) d DT where ReN = c N D 2i / c , and C10 is constant that should be determined from mass transfer data.
Di 0.592 1.328 KL,av 2C10 ReN . = √ −0.502 √ DT d DN
(87)
The experimental values of C10 have been tabulated in Table 4 over 120 runs for five different mass transfer systems analyzed by Lee (1978). The five systems studied are listed in Table 3. By averaging out the value of C10 from mass transfer data Eq. (87) changes to
KL,av 8.942×−7 −0.502 Di 0.592 1.328 ReN . d = √ √ DT DN
(88)
This correlation contains the least number of dimensionless groups and has an average absolute deviation of only 19.46%. √ The comparison of the experimental values of KL,av / DN with the proposed correlation is shown in Fig. 4 (Tables 3 and 4).
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B. Jajuee et al. / Chemical Engineering Science 61 (2006) 3917 – 3929
4. Conclusion A brief synopsis of the former mass transfer theories shows that the mathematical model developed in this paper is postulated closer to the truth than the conventional picture of an undisturbed layer at the surface of the fluid and of those theories lacking in turbulent or bulk motions. Nevertheless, it will never seem entirely self-evident unless the time-average overall mass transfer coefficient in mass transfer processes of practical interest approximates to Eq. (55) through qualitative observations on stirred and flowing fluids. In this regard, a vast body of literature dealing with bubbles and drops was reviewed to show that it is possible to shed new light on systems in which turbulent motion exists contemporaneously with bulk motion, while intensity of turbulence is the sequel to intensity of convection. However, the terminology of the stagnant film hypothesis would probably be retained as a matter of convenience in some cases.
x xA y z za zF
overall flux of species A relative to a phase boundary, mol/m2 s time-average overall flux of species A relative to a phase boundary, mol/m2 s average rate of exchange of the film surface, m2 /s Reynolds number, Dc Usg /
Reynolds number defined by Eq. (79) impeller Reynolds number, c N D 2i / c fractional rate of surface renewal, 1/s average rate of surface renewal, 1/s Schmidt number, D/ Schmidt number defined by Eq. (79) Sherwood number defined by Eq. (70) dimensionless time variable defined by Eq. (29) superficial gas velocity, m/s superficial liquid velocity, m/s molar average velocity; velocity scale of eddies in viscous dissipation range, m/s rectangular coordinate, m dimensionless mole fraction of species A rectangular coordinate, m rectangular coordinate, m distance from the interface to the bulk of fluid, m effective film thickness based of the film theory, m
Z ZA
dimensionless position variable defined by Eq. (28) dimensionless position variable defined by Eq. (32)
NA NA,av r Re Re ReN s sav Sc Sc Sh T Usg Usl v
Notation a A A0 A∞ Bo cA cA,i cA0 c∗ C Ci D Dc Di DT F Fr g G h H¯ JA kL K KL,av KL a l n N
specific interfacial surface, m2 /m3 exposed area of time-dependent surface, m2 initial area of the surface exposed to gas, m2 total area of the surface exposed to gas, m2 Bond number defined by Eq. (79) time-dependent molar concentration of species A, mol/m3 molar concentration of species A at the interface, mol/L3 molar concentration of species A at the bulk of fluid, mol/m3 saturation concentration, mol/m3 dimensionless concentration profile defined by Eq. (27) constant molecular diffusivity, m2 /s column diameter, m impeller diameter, m tank diameter, m defined by Eq. (34) Froude number defined by Eq. (79) gravitational acceleration, m/s2 defined by Eq. (35) defined by Eq. (44) average distance of approach of eddies to interface, m molar flux of diffusion of species A relative to the molar average velocity, mol/m2 s fluid-film mass transfer coefficient, m/s consistency index in a power-law model, kg sn−2 /m time-average overall mass transfer coefficient, m/s overall volumetric mass transfer coefficient, 1/s length scale of eddies in viscous dissipation range, m dimensionless exposed surface area defined in Eq. (54); flow index in a power-law model impeller speed, rps
Greek letters ∇ ˙ c 0 ϑ
shear rate, 1/s vector differential operator defined by Eq. (9) unit vector, m volume fraction, dimensionless energy dissipation rate per unit mass, W/kg time, s circulation time around vessel, s characteristic constant defined for each system; e.g. for gas–fluid contactors 0 might be bubble formation time, s amplitude of period, 1/s kinematic viscosity, m2 sn−2 viscosity, kg/m s density, kg/m3 interfacial tension, N/m dimensionless variable used in Eq. (54) dimensionless variable defined by Eq. (35) surface–age distribution function, m2 /s
Subscripts c BM d FM g p
continuous phase bubble mean (diameter) dispersed phase fluid particle mean (diameter) gas particles in general
B. Jajuee et al. / Chemical Engineering Science 61 (2006) 3917 – 3929
Acknowledgments A. Margaritis, D. Karamanev, and M.A. Bergougnou acknowledge financial support from Imperial Oil Ltd., Sarnia, Ontario, and Natural Sciences and Engineering Research Council (NSERC) of Canada. B. Jajuee acknowledges support by the Western Engineering Scholarships. References Akita, K., Yoshida, F., 1973. Gas holdup and volumetric mass transfer coefficient in bubble columns. Industrial & Engineering Chemistry Process Design and Development 12, 76–80. Angelo, J.B., Lightfoot, E.N., 1968. Mass transfer across mobile interfaces. A.I.Ch.E. Journal 14 (4), 531–540. Angelo, J.B., Lightfoot, E.N., Howard, D.W., 1966. Generalization of the penetration theory for surface stretch: application to forming and oscillation drops. A.I.Ch.E. Journal 12 (4), 751–760. Banerjee, S., Scott, D., Rhodes, E., 1968. Mass transfer to falling wavy liquid films in turbulent flow. Industrial & Engineering Chemistry Fundamentals 7 (1), 22–27. Calderbank, P.H., 1958. Physical rate processes in industrial fermentation, Part I: the interfacial area in gas–liquid contacting with mechanical agitation. Transactions of the Institution of Chemical Engineers 36, 443–463. Dankwerts, P.V., 1951. Significance of liquid-film coefficients in gas absorption. Industrial & Engineering Chemistry 43, 1460–1467. Deckwer, W.-D., Nguyen-Tien, K., Schumpe, A., Serpemen, Y., 1982. Oxygen mass transfer into aerated CMC solutions in a bubble column. Biotechnology and Bioengineering 24, 461–481. Dobbins, W.E., 1956. The Nature of the Oxygen Transfer Coefficient in Aeration Systems. Biological Treatment of Sewage and Industrial Wastes, vol. 1, pt. 2-1. Reinhold, New York, pp. 141–148. Durst, F., Tsiklauri, G.V., Afgan, N.H., 1979. Two-Phase Momentum, Heat and Mass Transfer in Chemical, Process, and Energy Engineering Systems, vol. 2, Washington, pp. 835–876 El. Temtamy, S.A., Khalil, S.A., Nour-El-Din, A.A., Gaber, A., 1984. Oxygen mass transfer in a bubble column bioreactor containing lysed yeast suspensions. Applied Microbiology and Biotechnology 19, 376–381. Godbole, S.P., Schumpe, A., Shah, Y.T., Carr, N.L., 1984. Hydrodynamics and mass transfer in non-Newtonian solutions in a bubble column. A.I.Ch.E. Journal 30 (2), 213–220. Harriott, P., 1962. A random eddy modification of the penetration theory. Chemical Engineering Science 17 (3), 149–154.
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Higbie, R., 1935. The rate of absorption of a pure gas into a still liquid during short periods of exposure. Transactions of the A.I.Ch.E. 31, 365–389. Jackson, M.L., Shen, C., 1978. Aeration and mixing in deep tank fermentation systems. A.I.Ch.E. Journal 24 (1), 63–71. Joseph, S., Shah, Y.T., Carr, N.L., 1984. Two bubble class model for mass transfer in a bubble column with internals. Institute of Chemical Engineers Symposium Series 87, 223–230. Kawase, Y., Halard, B., Moo-young, M., 1987. Theoretical prediction of volumetric mass transfer coefficients in bubble columns for Newtonian and non-Newtonian fluids. Chemical Engineering Science 42 (7), 1609–1617. King, C., 1966. Turbulent liquid phase mass transfer at a free gas–liquid interface. Industrial & Engineering Chemistry Fundamentals 5 (1), 1–8. Koide, K., Sato, H., Iwamoto, S., 1983. Gas holdup and volumetric liquidphase mass transfer coefficient in bubble column with draught tube and with gas dispersion into annulus. Journal of Chemical Engineering of Japan 16, 407–413. Kozinski, A.A., King, C.J., 1966. The influence of diffusivity on liquid phase mass transfer to the free interface in a stirred vessel. A.I.Ch.E. Journal 12 (1), 109–116. Lamont, J.C., Scott, D.S., 1970. An eddy cell model of mass transfer into the surface of a turbulent liquid. A.I.Ch.E. Journal 16 (4), 513–519. Lee, J.M., 1978. Mass transfer and liquid–liquid dispersion in agitated vessels. Ph.D. Dissertation, University of Kentucky, Lexington, KY. Lewis, W.K., Whitman, W.G., 1924. Principles of gas absorption. Industrial & Engineering Chemistry 16, 1215–1220. Nakanoh, M., Yoshida, F., 1980. Gas absorption by Newtonian and nonNewtonian liquids in a bubble column. Industrial & Engineering Chemistry Process Design and Development 19, 190–195. Nakao, K., Takeuchi, H., Kataoka, H., Kaji, H., Otake, T., Mlyauchi, T., 1983. Mass transfer characteristics of bubble columns in recirculation flow regime. Industrial & Engineering Chemistry Process Design and Development 22, 577–582. Nishikawa, M., Kato, H., Hashimoto, K., 1977. Heat transfer in aerated tower filled with non-Newtonian liquid. Industrial & Engineering Chemistry Process Design and Development 16 (1), 133–137. Shamlou, P.A., Pollard, D.J., Ison, A.P., 1995. Volumetric mass transfer coefficient in concentric-tube airlift bioreactors. Chemical Engineering Science 50 (10), 1579–1590. Skelland, A.H.P., Lee, L.M., 1981. Drop size and continuous-phase mass transfer in agitated vessels. A.I.Ch.E. Journal 27 (1), 99–111. Stewart, W.E., Angelo, J.B., Lightfoot, E.N., 1970. Forced convection in three-dimensional flows: II. A.I.Ch.E. Journal 16 (5), 771–786. Toor, H.L., Marchello, J.M., 1958. Film-penetration model for mass and heat transfer. A.I.Ch.E. Journal 4 (1), 97–101. Wang, T., Wang, J., Jin, Y., 2005. Theoretical prediction of flow regime transition in bubble columns by the population balance mode. Chemical Engineering Science 60, 6199–6209.
Application of surface-renewal-stretch model for interface mass transfer Babak Jajuee, Argyrios Margaritis ∗ , Dimitre Karamanev, Maurice A. Bergougnou Department of Chemical and Biochemical Engineering, Faculty of Engineering, University of Western Ontario, London, Ont., Canada N6A 5B9 Received 4 April 2005; received in revised form 8 December 2005; accepted 20 January 2006 Available online 9 March 2006
Abstract A new surface-renewal-stretch (SRS) model was developed to correlate experimental data for the time-average overall mass transfer coefficient, KL,av , in liquid–liquid and gas–liquid mass transfer systems. The model is based on the equation of continuity, which includes both turbulent and convective mass transfer at the liquid–liquid and gas–liquid interfaces. The model incorporates Dankwerts surface-renewal model with the penetration theory for surface stretch proposed by Angelo et al. [Angelo, J.B., Lightfoot, E.N., Howard, D.W., 1996. Generalization of the penetration theory for surface stretch: application to forming and oscillation drops. A.I.Ch.E. Journal 12 (4) 751–760]. We used our new SRS mass transfer model to correlate successfully the existing interface mass transfer experimental data from published literature. As a result, the experimental mass transfer coefficient data was predicted with a high degree of accuracy. 䉷 2006 Elsevier Ltd. All rights reserved. Keywords: Mass transfer coefficient; Interface; Bubble column; Gaslift contactor; Airflit bioreactor; Absorption; Bubble; Drop
1. Introduction Mass transfer theories are mainly developed for the process of absorption of a gas into a liquid, even though their application might be extended to the cases that mass transfer occurs between any two immiscible fluid phases. Before turning to the principles involved, the reader should be aware of certain terminology which is basic to understanding the material presented in this paper. For our purposes, a “fluid particle” is a self-contained body with maximum dimension between about 0.5 m and 10 cm, separated from the surrounding medium by a recognizable interface. The material forming the particle and the medium surrounding the particles will be termed the “dispersed phase” and the “bulk phase”, respectively. If the dispersed phase is in the liquid state, the particle is called a “drop”, and if it is a gas, the particle is referred to as a bubble. Together, drops and bubbles comprise “fluid particles”. Henceforth, we use “continuous phase” to refer to the “bulk phase” with convective motion. Moreover, consideration is limited in the first place to the cases in which the bulk phase and the ∗ Corresponding author. Tel.: +1 519 661 2146; fax: +1 519 661 4275.
E-mail address: [email protected] (A. Margaritis) URL: http://www.eng.uwo.ca/people/amargaritis/. 0009-2509/$ - see front matter 䉷 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2006.01.026
dispersed phase are both a Newtonian fluid and then it is extended to cases in which the bulk phase would be also non-Newtonian. The conventional and simplest picture of mass transfer between two fluid phases, borrowed from a similar concept used in the convective heat transfer, is that there exists a stagnant film at the interface (Lewis and Whitman, 1924). In this case, the surface concentration may be assumed to be the saturated ∗ , while the concentration of the bulk phase is concentration, cA kept uniform outside this film by turbulent mixing. The rate of mass transfer per unit area, also known as mass transfer flux, will then be constant (steady state) and given by the following expression: NA =
D (cA,i − cA0 ), zF
(1)
where NA is the net flux defined as the rate of absorption per unit area, cA,i is the concentration at the interface, cA0 is the bulk concentration, D is the molecular diffusivity and zF is the effective film thickness which depends upon the nature of the flow conditions. Therefore, the net flux appears to conform to the expression NA = kL (cA,i − cA0 ),
(2)
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B. Jajuee et al. / Chemical Engineering Science 61 (2006) 3917 – 3929
where kL is the film mass transfer coefficient and is constant for a fluid–fluid system under given conditions. Although it is preferable to refer to the conventional picture due to its simplicity as though it really exists, the fictitious nature of the film theory may lead to erroneous results in many actual mechanisms of mass transfer. In particular, when fluid particles rise or fall in infinite media with turbulent motion, the conditions required maintaining a stagnant film at the surface of moving fluid particles is lacking. In other words, it appears unlikely that the surface layer on each particle keeps its identity throughout all times. Higbie (1935) took a major step forward by introducing the “penetration theory”. This theory emphasizes that in many situations the time of exposure of a fluid to mass transfer is too short to let the concentration gradient of the film, characteristic of steady state, to develop. With the assumption of constant but short time of exposure for all fluid particles, they are subject to unsteady state diffusion or penetration of solute. Taking the uniform initial concentration of cA0 for the dissolved component and the surface concentration of cA,i , as soon as the bulk phase is exposed to the diffusing component, which may be also taken as the equilibrium solubility of it in the bulk of the fluid, the initial and boundary conditions become cA0 at = 0 for all z, (3) c = cA0 for all z = ∞, cA,i for all z = 0, in which is the time of exposure to the diffusing component. Thus, the average flux over the time of exposure is 4D NA,av () = (cA,i − cA0 ). (4) This theory pictures an unbroken film for component diffusion while the bulk phase is stagnant and has constant time of exposure for all eddies and fluid particles; nevertheless, the application of the penetration theory is probably extended to cases where the bulk phase is also in turbulent motion. It can be seen that the rate of absorption becomes very slow after a time unless the surface is renewed by stirring or by convection. This was first clarified by Dankwerts (1951) with “surface-renewal theory” as he pointed out that Higbie theory with its constant time of exposure of eddies at the surface is a special case of what may be a more realistic picture of the processes occurring during absorption into an agitated fluid where in fact eddies are exposed for varying lengths of time. In the surface-renewal theory, the interface is pictured then as a mosaic of different exposure-time surface elements where eddies are continually exposing fresh surfaces to the diffusing component and sweeping away and mixing into the bulk of the fluid. Dankwerts assumed that any portion of the bulk phase absorbs the component at a rate given by Eq. (4) in a case that the depth of penetration of the solute would be much smaller than the scale of turbulence, so that the intensity of turbulence accentuates velocity gradient existing beneath the surface and at the same time decreases the depth of penetration by shortening the period for any part of the bulk phase which is exposed to the diffusing
component before being submerged once again. To compensate for the lack of turbulence at the interface, Dankwerts added a new constant parameter called the mean rate of production of fresh surface, s, to the penetration theory and assumed that the chance of a surface element being replaced by another within a given time is independent of how long it has been in the surface; hence the fractional rate of replacement of the elements belonging to different age groups is equal to s. Combination of the continuity equation with the same conditions of Eq. (3) and the concept of surface renewal gives a new expression to the mean rate of mass transfer per unit area of turbulent surface √ (5) NA,av = Ds(cA,i − cA0 ). The penetration and surface-renewal theories tacitly assume that immediately underlying all freshly formed surfaces is fluid with the same concentration, cA0 , of dissolved component. It is equal to consider the surface elements to be infinitely deep that diffusing component never reaches the region of constant concentration below. Dobbins (1956) pointed out that this is not true for aeration systems where flowing streams continually absorb oxygen and the concentration of the bulk of the fluid changes. Accordingly, Dobbins considered a finite depth of surface elements and eddies by replacing the third boundary condition in Eq. (3) with cA = cA0 for z = za and obtained ⎡ ⎤ 2 √ sz a⎦ (6) (cA,i − cA0 ). NA,av = ⎣ Ds coth D n , with n It can be seen that the general dependence, KL ∝ DAB dependent upon circumstances, is satisfied in this expression. Other researchers have made similar suggestions (King, 1966; Kozinski and King, 1966; Toor and Marchello, 1958; Lamont and Scott, 1970). In summary, for rapid penetration (D), or for thin surface elements, and small rate of surface renewal (s), the situation is close to steady state and the predominant character of the mass transfer is described by the film theory; whereas for slow penetration, or for short time of exposure or rapid renewal (unsteady state situation) it follows Eqs. (5) or (6). Remarkably, Eq. (5) may be applied for bulk phase moving parallel to the surface with a velocity that varies with the depth while the time of exposure is so short that the depth of penetration is much smaller than the depth at which the velocity is appreciably different from that at the surface. As a complimentary to this, Eq. (6) depicts a more realistic picture of the penetration depth by assuming a finite depth of surface element for fluid particles. Molecular diffusivity is the most important attribute in concentration changes among unsteady state theories discussed above. In addition to transport by molecular motion, however, mass may also be transported by the bulk motion. Therefore, a comprehensive net flux of mass transfer in the liquid including bulk motion is the combined net flux vector, which consists of the both molecular flux vector and convective flux vector
N A = JA + cA v,
(7)
B. Jajuee et al. / Chemical Engineering Science 61 (2006) 3917 – 3929
where JA and v are the molar molecular flux vector and the molar average velocity vector, respectively. A, JA = −cD ∇x v =
N
xA vA ,
(8)
A=1
where ∇ is the vector differential operator and is defined in rectangular coordinates as ∇ = x
j j j + y + z jx jy jz
(9)
in which is the unit vector. Angelo and coworkers (Angelo et al., 1966; Angelo and Lightfoot, 1968; Stewart et al., 1970) developed the penetration concepts in the absence of turbulence when there is bulk motion by introducing a promising theory called “surface-stretch theory”. On the assumption that diffusion happens in the same direction as convection, i.e., z, the surface-stretch theory expresses the local velocity vz in time-dependent interface surface A, and gives the continuity equation a new form −z
j ln A j
jcA jcA j 2 cA + =D 2 j jz x,y jz
(10)
whose solution leads to another expression for the net flux ⎤ ⎡ ⎢ (A(/0 )) D/0 ⎥ NA,av = ⎣ (11) ⎦ (cA,i − cA0 ), /0 2 A () d 0 where A is the interfacial surface through which mass transfer takes place while it changes periodically with time, and 0 is a constant with the dimension of time defined for each system. Eq. (10) will be justified in the discussion of surface-renewalstretch theory. In brief, the concepts of bulk motion and turbulent motion have been neglected in the surface-renewal and surface-stretch theories, respectively. Presented here is a holistic approach to the treatment of mass transfer processes occurring in fluid–fluid contactors, while the symbiotic relationship exist between turbulent and bulk motions synchronously. 2. Fluid–fluid contactors More recently, fluid–fluid contactors are referred to a myriad of configurations developed for two or three-phase applications in which fluid particles, e.g. bubbles or drops, have interactions. Fluid particles oscillate and wobble when rising through a denser media and make surfaces wavy and rippled. Gaslift systems considered as gas–liquid contactors, for example, are widely used in many operations such as fermentation and wastewater treatment especially where microorganisms are used as biocatalysts. In general, they are pictured as pneumatically agitated devices characterized by circulation in a defined cyclic pattern where slurry or substrate circulates from the riser
3919
to the downcomer section. Aeration of one section exploits hydrostatic pressure differences by making lower hydrostatic head compared to the non-aerated section at the same vertical coordinate across the membrane wall (King, 1966). Unsteady circulation with rocking motion of fluid particles increases the intensity of turbulence, which in turn changes incessantly the conformation, and position of eddies and exposes fresh surfaces to fluid particles. At the same time, bulk motion also drives eddies and liquid particles in a continuous circulation. It is well known that the mass transfer processes occurring in fluid–fluid contactors are too complicated to be explained by an implausible theory. On the other hand, providing a thorough picture of mass transfer, for many purposes, is impossible without taking the inevitable existence of both bulk and turbulent motions into account. 3. Theoretical development 3.1. Surface-renewal-stretch (SRS) model The main objective of this work is to extend the penetration theory of mass transfer to fluid–fluid interacting systems, e.g. gaslift contactors, in which both turbulent and bulk motions exist. Consider a bulk phase with uniform rate of mass transfer through the interface that is maintained in turbulent and bulk motion, as shown in Fig. 1. This case may be expounded from both macroscopic and microscopic standpoints. The former is pictured with eddies and fluid particles that have an explicit bulk motion, while the latter with the surface of each element that is continuously replaced with fresh surface and has a distribution of ages varying from zero to infinity at any particular instance. Let the total area of the surface exposed to the gas would be equal to A∞ . On the assumption that the chance of any portion of surface element being replaced by another is quite independent of how long it has been in the surface, the average rate of exchange of the film which produces fresh surfaces is taken to be constant and equal to r(). The concept of surface–age distribution function might be borrowed from the surface-renewal theory, in the sense that at any moment the distribution of ages of the film area is given by the function (), and the film area comprising the elements having ages between and + d is () d. This does not vary with time at steady state and is equal to the film area in the age group between − d and less the portion of this, which is replaced by fresh surface in a short time interval equal to d
r () d = ( − d) d 1 − d . (12) A∞ Writing Taylor series for ( − d) gives () = () −
r d() () d. d − d A∞
(13)
Thus r d() =− () d A∞
(14)
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B. Jajuee et al. / Chemical Engineering Science 61 (2006) 3917 – 3929 vz
Distance, z
Finite depth of element surface
cA0 For all
za At z=0 for all > 0 cA,i
Concentration
Fluid Particle
Fig. 1. Mass transfer mechanism in a fluid–fluid contactor when the scale of turbulence and convection are both taken into account.
and since ∞ () d = A∞
(15)
0
the distribution function can be obtained as () = re−s ,
(16)
where s has the value of r/A∞ , defined as the rate of production of fresh surface which is exposed to the gas (Dankwerts, 1951). The area of the mass transfer surface, () d, is a function of time and may be readily integrated to yield A() = −A∞ e−s + C1 .
(17)
Taking A = A0 at = 0 gives A() = A∞ (1 − e−s ) + A0 .
(18)
This expression represents the area of the mass transfer surface changes with time and the intensity of turbulence, s. On the other hand, this area may also change with time due to convection, which produces turbulent motion in turn. We limit consideration to situations in which diffusion happens in the direction of convection, perpendicular to the interfacial surface area. We shall also assume that at any given instant an interface, which expands and shrinks with time, exists between the continuous fluid and fluid particles rising in the z direction with the bulk motion, as shown in Fig. 1. Under these circumstances, diffusion is negligible in directions parallel to the interface and is also considered one dimensional in the neighborhood of any surface element. The continuity equation, therefore, takes the form vz
jcA jcA j 2 cA + =D 2 , jz j jz
(19)
where vz may be derived by making an overall momentum balance over each contactor system; however, it is also attributable to the corresponding interfacial area of the time-dependent surface. In the case of constant molar density of the solution, consider a small flat element of surface xy moving with the fluid bulk velocity. The direction of the z-axis is pointing into the diffusing phase whose net rate of mass transfer is assumed to be small. The area of the surface of this element changes with time
jA = [x(vy |y+y − vy |y ) + y(vx |x+x − vx |x )]. j ave (20) For any point in the interfacial surface, we may divide Eq. (20) by xy and take the limit as x, y approach zero
jvy j ln A jvx = + . (21) j jx jy When the density is constant this expression may be written with the aid of the equation of continuity as
j ln A jvz =− . (22) j jz Since the expression on the left-hand side is independent of z, we may write for negligible net transfer across the interface
j ln A vz = −z , (23) j where the overall interfacial area A() is obtainable from Eq. (18) at any time. It is worth noting that z should be small compared to the local radii of curvature for curved surfaces like the interfacial surface of a gas bubble. This is consistent with the limitations on the penetration theory. Eqs. (19) and (23) are
B. Jajuee et al. / Chemical Engineering Science 61 (2006) 3917 – 3929
then combined to give Eq. (10) −z
j ln A j
jcA jcA j 2 cA + =D 2 . j jz x,y jz
(10)
I.C. c = cA0
at = 0 for all z,
(24)
B.C.1
c = cA0
for all at z = za ,
(25)
B.C.2
c = cA,i
for all at z = 0.
(26)
The initial condition and the first boundary condition above follow the assumed uniform bulk concentration of gas in entire solution and finite depth of the surface elements or eddies. Nevertheless, the rough assumption of infinitely deep surface element is constantly referred to as though there are short constant times. This may be regarded as a harmless and convenient usage for many systems but not for aerated flowing systems in which bulk concentration continually changes with time. In airlift systems, the dissolving solute is able to reach the depth za corresponding to the thickness of the eddy, so that from the solute point of view, za is essentially finite. The second boundary condition expresses the concentration in the continuous phase at the interface, cA,i , which may be taken as the equilibrium solubility of the fluid particles in the fluid. The concept of each concentration has been illustrated schematically in Fig 1. To solve Eq. (10) we may define the following dimensionless variables: C=
cA − cA,i , cA0 − cA,i
Eq. (30) is a partial differential equation whose easiest way of solving is considering a combination of variables in order to convert it to a simple differential equation C = C(),
For simplicity sake, we shall assume that D would be constant. The boundary conditions over Eq. (10) then are
(27)
3921
= Z × G(T ).
Hence, Eq. (30) becomes F G dC d2 C + − 3 = 0, G2 G d d2
(35)
(36)
where G indicates the derivative of G with respect to T at constant Z. The principle of “combination of variables” requires G F = , (37) − G2 G3 where is an arbitrary constant. Eq. (36) is thus simplified to d2 C dC + = 0. 2 d d
(38)
In brief, Eq. (30) was reduced to Eq. (38), which is an ordinary differential equation and has the solution of √ √ 2 C() = C2 + C3 × erf . (39) 2 If the assumed combination of variables is successful it should be necessary that Eq. (39) satisfy all Eqs. (31)–(33). Application of the initial condition and the two boundary conditions permits the evaluation of C2 and C3 which are constant C(0) = 0,
(40)
C(A ) = 1,
(41)
z Z= √ , D0
(28)
where A is the value of when Z = ZA . Eq. (37) may be rearranged more simply as
, 0
(29)
F G − G = G3 .
T =
where 0 is a constant with units of time and is evaluated with time characteristic of each system; e.g. for a gaslift contactor 0 might be bubble-formation time. The dimensionless form of our system becomes jC jC j2 C , − F (T )Z = jT jZ jZ 2
(30)
C = 1 at T = 0 for all Z,
(31)
C=1 C=0
for all T at Z = ZA = √ for all T at Z = 0,
za , D0
F (T ) =
j ln A(T ) jT
C() =
erf() . erf(A )
(43)
Eq. (42) is a Bernoulli equation and, therefore, by the substitution G = h−1/2
(44)
and the integrating factor (33)
e2
F (T ) dT
= e2 (j ln A/jT ) dT = A(T )2
(45)
it takes the form
. x,y
If the chosen combination of variables is to be successful, Eqs. (40) and (41) represent restrictions that should be met. For convenience we take = 2, thereby giving
(32)
where
(42)
(34)
A2
dh dA = −2hA + 4A2 . dT dT
(46)
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B. Jajuee et al. / Chemical Engineering Science 61 (2006) 3917 – 3929
We may readily solve Eq. (46) whereby it follows that A(T ) , G(T ) = T 4 0 A2 () d + C4
(47)
where C4 is a constant that must be zero to satisfy Eq. (41). Thus, A(T ) . G(T ) = T 2 4 0 A () d
(48)
To obtain the complete solution of Eq. (30) we should recall that = Z × G(T )
T 2 erf Z × A(T ) 4 0 A () d
, C= (49) T 2 erf ZA × A(T ) 4 0 A () d where ZA is a constant regarding to za in which C =1. This expression satisfies Eqs. (32)–(33). It also automatically satisfies Eq. (31) if one substitutes A(T ) from Eq. (18) into Eq. (49). Returning to the concept of age–surface distribution function, the instantaneous rate of absorption for surface elements having age and film area () d is (T ) dT , dNA,z = jA,z ∞ 0 (T ) dT where jA,z = −D hence
(50)
jZ jcA jC = (c − c )D A,i A0 jz z=0 jZ Z=0 jz
(51)
Ds 2 0 dNA,z = (cA,i − cA0 ) ⎡ ⎤ T 2 () d A(T ) A 0 ⎢ ⎥
⎥ ×⎢ ⎣ ⎦ T erf ZA A(T ) 4 0 A2 () d × exp((−s0 )T ) dT = (cA,i − cA0 )KL (T ) dT ,
(52)
where KL is the local mass transfer coefficient at a given time on the interface. Hence the average flux over the time of exposure per unit area of turbulent surface is ∞ NA,av = (cA,i − cA0 ) KL (T ) dT 0
= KL,av (cA,i − cA0 ).
(53)
where n is a constant equal to the ratio of A0 /A∞ which is essentially less than unity and greater than zero, and is a dimensionless variable taken as equal to s0 T . The price that is paid for a greater accuracy in analytical models is an increase in complexity. The above analysis represents a comprehensive expression for the average mass transfer coefficient in any fluid–fluid system in which turbulent motion, bulk motion, and a finite depth of penetration exist. All previous mass transfer theories discussed earlier can be developed as special cases of the above treatment. It can be seen that the overall mass transfer coefficient is a function of the diffusion coefficient, D, the fractional rate of surface renewal, s, and the depth of penetration, za . The main drawback to such a complicated expression for mass transfer coefficient is having an error function in denominator which makes the integral of Eq. (54) too convoluted to be employed for practical purposes; nevertheless, in many cases some assumptions help to circumvent the difficulty of the problem. When the scale of turbulence is high, as in the case of many practical systems, the rate of production of fresh surface increases and produces infinitely deep liquid elements corresponding to the thickness of eddies due to the short exposure times. Hence za approaches infinity and the error function becomes close to unity. Moreover, regarding the fact that 0 < n < 1, the integrand in Eq. (54) finds virtually the value of 2 (Maple, 2003) when the error function is equal to unity, and thus 4Ds KL,av = . (55) This is a new model of time-average overall mass transfer coefficient coming out of the equation of continuity in the case that the two immiscible fluid phases undergo turbulent motion with one-dimensional diffusion and convective motion in the same direction. It is noteworthy to recall that Eq. (49) was achieved by ascribing change of time-dependent exposed surface area and increase of intensity of turbulence to convection motion, as expressed in Eq. (23), where vz is convective molar average velocity. As a result, s is a representative of the scale of turbulence vis-à-vis convective motion that renews the surface of the liquid by either stirring or convection, and hence Eq. (55) embodies the influence of both turbulent and convective motions. In a special case that mass transfer takes place into a stagnant fluid of infinite depth, Eq. (55) changes to the model of penetration theory. The time-average overall mass transfer flux becomes 4Ds NA,av = (cA,i − cA0 ). (56) Provided no chemical reaction occurs, a different significance is merely attached to the overall mass transfer coefficient while all expressions derived for the rate of mass transfer are similar to that based on the film theory. For systems of practical
To solve Eq. (53) through Eq. (52), A(T ) should be substituted from Eq. (18) into Eq. (52) and integrated when T changes from zero to infinity. − + (1 + n))e− ∞ (−e −0.5e−2 + 2(1 + n)e− + (1 + n)2 − 1.5 − 2n Ds d, KL,av = 0 erf (s × z2 )/4D(−e− + (1 + n)) −0.5e−2 + 2(1 + n)e− + (1 + n)2 − 1.5 − 2n a
(54)
B. Jajuee et al. / Chemical Engineering Science 61 (2006) 3917 – 3929
3923
importance, average mass transfer coefficients can be evaluated experimentally in such an easy way without being concerned about parameters like D, s, za , if the average flux is obtained from the general expression of Eq. (53). Having selected an appropriate system, the equation of continuity for species A in a binary system is written over the volume of a shell in the form
than 0.5, where H is the average distance to where eddies approach the interface. In this case, the length and velocity scales of the eddies for isotropic turbulence can be written as 3 1/4 l= , v = ( ˙)1/4 , (61) ˙
dNA,z dc + (1 − p ) = 0. dz d
where = / is kinematic viscosity and ˙ is viscous dissipation of energy per unit mass of eddies. The connotation of Kolmogoroffs’ time scale of eddies in viscous dissipation range can be used in the jargon of surface-renewal theory, in the sense that s is defined (Banerjee et al., 1968)
(57)
It should be noted that Eq. (57) is only valid when absorption takes place in the z direction with no chemical reaction. Assuming jz ≈ z = a −1 , it may be written at any given instant as −KL a(cA,i
dc − c) + (1 − p ) = 0, d
(58)
where a is the specific surface area, i.e., the interfacial surface area divided by the bulk volume of fluid–fluid contactors. The negative sign of the expression on the left-hand side indicates that the net rate of mass transfer changes inversely with depth. We may readily solve Eq. (58) by rearranging and getting an integral from zero to time c KL a 0 d dc = (59) (1 − p ) c0 (cA,i − c) then, ln
cA,i − c (−KL a) = . cA,i − cA0 (1 − p )
(60)
The experimental data might be obtained to evaluate KL a from Eq. (60), while finding a good correlation for s as a function of measurable parameters, e.g. fluid–fluid skip velocity, superficial velocities of two fluid phases, and diameter of fluid particles, provides a good picture based on the surface renewalstretch model to predict the behavior of fluid–fluid contactors in different situations. On the evidence of literature, Eq. (60) seems virtually ambiguous by being directly derived from the film theory. This happens because the net flux of mass transfer in the film theory also follows the same general expression as in Eq. (53). The accuracy of Eq. (55) can be easily scrutinized by analyzing the effects of the scales of turbulence and convection motion on the overall mass transfer coefficient. 3.2. Mass transfer coefficient correlation for gas bubbles Consideration is limited in the first place to the cases in which the dispersed phase is in the gas state. Due to its great acceptance, the determination of the mean fractional rate of surface renewal, s, through physical and hydrodynamic arguments has been the subject of much research interest (Banerjee et al., 1968; Skelland and Lee, 1981). In this work, the value of the mean rate of production of fresh surface has been determined without recourse to observations of mass transfer rates, as used in previous studies (Banerjee et al., 1968). Harriott (1962) showed that the concept of surface-renewal the√ ory could only be expected when the group H / D/s is less
s≡
v . l
Substitution of Eq. (61) for v and l into Eq. (62) yields ˙ s= .
(62)
(63)
Energy dissipation rate is calculated as follows (Wang et al., 2005):
g ˙ = Usg − (64a) Usl g. 1 − g In bubble columns and gaslift systems with low superficial liquid velocity (Usl ≈ 0) the energy dissipation rate can be approximated as ˙ = Usg g.
(64b)
Combination of Eqs. (63) and (55) provides an expression for s, and hence the time-average overall mass transfer coefficient becomes 1/2 4D Usg g . (65) KL,av = In deriving this expression it has been assumed that the contacting phases are Newtonian fluids with gas bubbles composing the dispersed phase and that the rate of surface renewal and the rate of energy dissipation follow the Eqs. (62) and (64b), respectively. The latter is almost certainly an oversimplification, but the approximations are probably no more drastic than that involved in real cases of mass transfer. Reviewing the literature gives a general indication of the behavior which is expected from this type of systems. Representing the non-Newtonian behavior of fluids by the power-law model, = K n leads to the expression 1/2
4D Usg g 1/(1+n) . KL,av = Similarly the analogs of Eq. (61) are 3 1/2(1+n) , v = ( ˙n )1/2(1+n) l = 2−n ˙
(66)
(67)
(68)
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B. Jajuee et al. / Chemical Engineering Science 61 (2006) 3917 – 3929
Table 1 A summary of investigator designs and operational variables for Fig. 2 Reference
Liquid
Dc (m)
× 103 0.92
991 995 1030 1130 1230
(Pa s)
(kg/m3 )
× 102
C8
Ave. Abs. Dev. from Eq. (80) (%)
7.02
1.326 1.571
15.2 28.04
7.1 7.1 7.1 7.1
0.851 1.044 8.13 0.512
21.19 1.65 26.1 50.44
7.26
0.52
22.83
(N/m)
Kawase et al. (1987)
Water
0.23 0.76
Nakanoh and Yoshida (1980)
Water 10% sucrose 30% sucrose 50% sucrose
0.1455
0.80 1.0815 2.543 10.85
Shamlou et al. (1995)
Fermentation Broth
0.317
0.894
El. Temtamy et al. (1984)
Water
0.15
1
1000
7.26
0.687
23.1
Koide et al. (1983)
Water 100 mol/m3 BaCl2 50% glycerol
0.14
0.8941 0.9185 5.93
997 1016 1141
7.196 7.242 6.688
0.682 0.846 1.496
57.1 23.73 30.09
Nakao et al. (1983)
Water
0.6
1.73
998.5
7.39
1.838
37.09
Jackson and Shen (1978)
Water
1.83 7.62
1.224 1.224
999.4 999.4
7.45 7.45
1.16 0.959
12.1 7.2
in which = K/ . Moreover, in both Newtonian and nonNewtonian solutions the average shear rate varies about proportional to superficial gas velocity (Nishikawa et al., 1977) = 5000Usg .
(69)
This has been regarded, for many purposes, as a useful relationship for derivation of mass transfer correlations (Nakanoh and Yoshida, 1980; Deckwer et al., 1982, Godbole et al., 1984). If unit volume of a two-immiscible-fluid mixture contains a dispersed phase volume of p made up of n fluid particles of diameter dp , then n = p /(dp3 /6). If the interfacial area in the unit volume is a, then n = a/dp2 . Equating the two expressions for n provides the specific area a=
6p dp
(70)
Similar correlations have been proposed by other researches in which the range of the exponent varies from 0.4 to 1 (Durst et al., 1979). Hence,
˙0.4 0.6 1− d 0.25 . a = C7 0.6 p
c
(74)
In the case of gas bubbles, Godbole et al. (1984) proposed the following correlation for gas hold up in non-Newtonian solutions: 0.6 −0.19
eff , g = 0.255Usg
(75)
where eff is defined as follows:
eff = K n−1 .
(76)
Substituting Eqs. (69) and (76) into Eq. (75) yields
and
(0.79−0.19n) . g = (0.255 × 50000.19(1−n) )K −0.19 Usg
6g a= dBM
(77)
(71)
when gas bubbles comprise the dispersed phase. For noncoalescing conditions, the size of gas bubbles produced in gas–liquid dispersion has been determined from a balance between surface tension forces and those due turbulent fluctuations (Calderbank, 1958) dBM = C5
997.1
0.6 ˙0.4 0.6
.
(72)
As a result Calderbank (1958) proposed the following expression for fluid particle size, dF M :
0.6 0.25 dF M = C6 0.4 0.6 p d . (73)
c ˙
This correlation agrees reasonably well with a vast body of literature data dealing with bubbles (Kawase et al., 1987). Assuming = 21 in Eq. (74) and combining Eqs. (64), (74), and (77) gives a new expression for specific gas–liquid interfacial area (8−n)/10
a = (0.5 × 50001−n/10 )C7
g 0.4 0.6 Usg
0.6 K 0.1
d
c
0.25 .
(78)
Combining Eqs. (78) and (67) and rearranging into dimensionless groups lead to a correlation for the volumetric mass transfer coefficient C8 0.5 (4n+17.5)/10(n+2) Sh = 5000(1−n)/10 √ Sc Re 0.6 (5n−2)/15(n+1) × Bo F r ,
(79)
B. Jajuee et al. / Chemical Engineering Science 61 (2006) 3917 – 3929
3925
Table 2 A summary of investigator designs and operational variables for Fig. 3 Reference
Liquid
n
Dc (m)
K (Pa s)
Kawase et al. (1987)
Carbopol-1 CMC-2
0.82 0.476
0.23
2.78 × 10−2 2.32
Nakanoh and Yoshida (1980)
0.3% CMC 0.5% CMC 1.0% CMC
0.97 0.985 0.835
0.1455
Godbole et al. (1984)
CMC-1 CMC-2 CMC-3 CMC-4 CMC-5
0.697 0.654 0.668 0.607 0.492
0.305
El. Temtamy et al. (1984)
Yeast-B 0.35% CMC
0.82 0.59
0.15 0.305
Joseph et al. (1984)
3% CMC
0.5242
× 102
(kg/m3 )
C8
Ave. Abs. Dev. from Eq. (79) (%)
(N/m)
992 992
6.92 7.11
0.522 0.109
3.16 17.37
1.15 1.626 12.11
1000 1000 1001
7.1 7.1 6.84
0.679 0.696 0.356
37.31 40.31 59.46
0.095 0.184 0.256 0.526 2.816
1000 1000 1002 1003 1005
7.3 7.25 7.32 6.84 6.77
0.492 0.349 0.262 0.192 0.086
27.33 20.65 3.41 3.47 18.19
1006 1002
9.67 7.10
0.624 0.189
12.37 4.62
1004
7.12
0.127
3.07
9 × 10−3 0.947 1.434
108
Shamlou et al. (1995) Kawase et al. (1987), Dc=0.76m Kawase et al. (1987), Dc=0.23m
5%
-2
El. Temtamy et al. (1984) Koide et al., (1983) Nakao et al. (1983) Nakanoh & Yoshida (1980) Jackson & Shen (1978), Dc=7.62m Jackson & Shen (1978), Dc=1.83m
7
Shexp
10
5%
+2
106
105 5x104 5x104
105
106
Shcalc
107
108
Fig. 2. Experimental Sherwood number for Newtonian fluids vs. predicted Sherwood number from Eq. (80).
where Sh = Bo =
KL aD 2c D gD 2c
,
Sc =
K/ Dc1−n , DU 1−n sg 2 Usg
and F r =
Dc g
Re =
2−n Dcn Usg
K/
,
.
For Newtonian fluids where n = 1, Eq. (79) reduces to C Sh = √8 Sc0.5 Re0.72 Bo0.6 Fr0.1 ,
(80)
in which C8 is the value of C8 when n in Eq. (79) is equal to unity. The exponential dependency of KL a on Usg is 0.95. It can be seen that the volumetric mass transfer coefficient is a
function of a number of variables including the superficial gas velocity, the dispersed and continuous phase properties, and the bubble size distribution. To keep our scope sufficiently broad, a comprehensive review of literature abounded with mass transfer data for both Newtonian and non-Newtonian fluids has been presented in Tables 1 and 2, respectively. The values of C8 have been determined for each set of mass transfer data through lack of reliable data for specific gas–liquid interfacial area. Accordingly, the constant C8 can be iteratively approximated as C8 = 1.022n3.3 .
(81)
If the value of the integrand in Eq. (54) were to be exactly equal to 2 and no assumption was made to obtain Eq. (80),
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B. Jajuee et al. / Chemical Engineering Science 61 (2006) 3917 – 3929
107 Kawase et al. (1987) Nakanoh & Yoshida (1980) Godbole et al. (1984) El. Temtamy et al. (1984) Joseph et al. (1984)
9% -1 9% +1
Shexp
106
105
104 104
105
106
107
Shcalc Fig. 3. Experimental Sherwood number for non-Newtonian fluids vs. predicted Sherwood number from Eq. (79).
i.e., the one in Eq. (64b), then the coefficient on the righthand side of Eq. (81) would be expected to be 1 rather than 1.022. The proposed correlation, Eq. (80), and the experimental results obtained by various workers for Newtonian fluids are compared in Fig. 2. The upper and lower dashed lines represent the negative and positive deviations from Eq. (80). The results are in satisfactory agreement over the wide range of 0.14mDc 7.62m with the average absolute deviation of only 25%. Absolute deviation is defined as |(Calculated value − Experimental value)/Experimental value| throughout this study. A very similar expression was introduced by Kawase et al. (1987) 0.8 Sh = √ Sc0.5 Re0.75 Bo0.6 Fr0.12 . (82) Considering Higbie’s penetration theory as a particular form of Eq. (55) and the theoretical basis for derivation of Eq. (82) is deemed to be the main reason for this close resemblance. Eq. (82) and other similar correlations (Nakanoh and Yoshida, 1980; Akita and Yoshida, 1973) approve the plausible picture of the model proposed in this work. Fig. 3 is also a plot of the predicted values of Sherwood number for non-Newtonian vs. those observed experimentally. Even though the exponential dependency of KL a on Usg , Dc , D, K/ , and is quite more complicated for non-Newtonian fluids, it provides much lower average absolute deviation of about 19% compared to previous published correlations (Kawase et al., 1987). Figs. 2 and 3 show how the points are evenly distributed around the diagonal.
for which the surface renewal rate would be determined in terms of hydrodynamic parameters, is sparse. Skelland and Lee (1981) suggested a periodically varying rate of surface renewal, reasoning that the condition required to maintain equal degree of turbulence at various locations in vessels for circulating drops of the dispersed phase appears to be lacking. In other words, the degree of turbulence changes and finds its maximum value near the impeller, minimum in the corners, and of intermediate strength in regions between these locations. The proposed relationship for the varying rate of surface renewal is s = sav + ϑ sin
2 , c
(83)
where c is the average circulation time of the droplets around the vessel. This concept is not obviously in harmony with Dankwerts (1951) theory with its constant surface-renewal rate and the one developed in this work. Substituting Eq. (83) into Eq. (14) yields
ϑc 2 () = C9 exp −sav + , (84) cos 2 c Table 3 The specifications of five systems studied for drops System
Dispersed phase
Continuous phase
Solute
1 2
Dimethyl siloxane Dimethyl siloxane
Heptanoic acid Heptanoic acid
3.3. Mass transfer coefficient correlation for drops
3
Dimethyl siloxane
To date, we only considered the cases in which the dispersed phase is a gas. The information on mass transfer between drops,
4 5
Ethyl acetate Benzaldehyde
Distilled Distilled Colonial Distilled Colonial Distilled Distilled
water water +20% pure cane sugar water +30% pure cane sugar water water
Heptanoic acid Heptanoic acid Heptanoic acid
B. Jajuee et al. / Chemical Engineering Science 61 (2006) 3917 – 3929
3927
10 0% -2
System 1. System 2.
0% +2
System 3. System 4.
KL,av /(ND) 0.5Exp
System 5.
1
1
10
KL,av /(ND)0.5Calc √ √ Fig. 4. Experimental KL,av / N D vs. predicted KL,av / N D from Eq. (88). Table 4 A summary of investigator designs and operational variables for Fig. 4 System
1 2 3 4 5
c × 103
d × 103
(Pa s) 0.0010 0.0018 0.0029 0.0010 0.0010
c
d
(Pa s)
(kg/m3 )
(kg/m3 )
0.0019 0.0019 0.0019 0.00046 0.0014
1000 1087 1131 1000 1000
873 873 873 894 1041
× 102
C10
(N/m)
D × 1010 (m2 /s)
Ave. Abs. Dev. from Eq. (88) (%)
0.039 0.032 0.033 0.006 0.015
6.01 5.66 4.02 6.01 6.01
4.857 × 10−7 3.190 × 10−7 4.399 × 10−7 7.037 × 10−7 7.715 × 10−7
16.72 43.24 12.50 35.95 40.14
Internal diameters of vessels: 0.21 m and 0.246 m. Diameters of impellers: 0.076 m and 0.106 m.
where C9 is obtainable from Eq. (15). Thus,
∞ A∞ ϑc 2 = exp −sav + d. cos C9 2 c 0
Taking s ≈ sav and substituting directly Eq. (86) into Eq. (55) yields (85)
This integral has been calculated and employed to evaluate the mass transfer coefficient for agitated vessels without bulk motion (Lee, 1978). In the case of agitation accompanied with bulk motion, A() should be evaluated afresh from Eqs. (84), substituted in Eq. (52), and finally integrated from zero to infinity to evaluate the time-average overall mass transfer coefficient; as presented in the case of constant surface-renewal rate in this paper. For the case that H = DT , a general correlation for sav obtained from statistical analysis method has been developed by Skelland and Lee (1981) as follows:
√ Di 0.592 1.328 0.5 sav = C10 −0.502 ReN N , (86) d DT where ReN = c N D 2i / c , and C10 is constant that should be determined from mass transfer data.
Di 0.592 1.328 KL,av 2C10 ReN . = √ −0.502 √ DT d DN
(87)
The experimental values of C10 have been tabulated in Table 4 over 120 runs for five different mass transfer systems analyzed by Lee (1978). The five systems studied are listed in Table 3. By averaging out the value of C10 from mass transfer data Eq. (87) changes to
KL,av 8.942×−7 −0.502 Di 0.592 1.328 ReN . d = √ √ DT DN
(88)
This correlation contains the least number of dimensionless groups and has an average absolute deviation of only 19.46%. √ The comparison of the experimental values of KL,av / DN with the proposed correlation is shown in Fig. 4 (Tables 3 and 4).
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B. Jajuee et al. / Chemical Engineering Science 61 (2006) 3917 – 3929
4. Conclusion A brief synopsis of the former mass transfer theories shows that the mathematical model developed in this paper is postulated closer to the truth than the conventional picture of an undisturbed layer at the surface of the fluid and of those theories lacking in turbulent or bulk motions. Nevertheless, it will never seem entirely self-evident unless the time-average overall mass transfer coefficient in mass transfer processes of practical interest approximates to Eq. (55) through qualitative observations on stirred and flowing fluids. In this regard, a vast body of literature dealing with bubbles and drops was reviewed to show that it is possible to shed new light on systems in which turbulent motion exists contemporaneously with bulk motion, while intensity of turbulence is the sequel to intensity of convection. However, the terminology of the stagnant film hypothesis would probably be retained as a matter of convenience in some cases.
x xA y z za zF
overall flux of species A relative to a phase boundary, mol/m2 s time-average overall flux of species A relative to a phase boundary, mol/m2 s average rate of exchange of the film surface, m2 /s Reynolds number, Dc Usg /
Reynolds number defined by Eq. (79) impeller Reynolds number, c N D 2i / c fractional rate of surface renewal, 1/s average rate of surface renewal, 1/s Schmidt number, D/ Schmidt number defined by Eq. (79) Sherwood number defined by Eq. (70) dimensionless time variable defined by Eq. (29) superficial gas velocity, m/s superficial liquid velocity, m/s molar average velocity; velocity scale of eddies in viscous dissipation range, m/s rectangular coordinate, m dimensionless mole fraction of species A rectangular coordinate, m rectangular coordinate, m distance from the interface to the bulk of fluid, m effective film thickness based of the film theory, m
Z ZA
dimensionless position variable defined by Eq. (28) dimensionless position variable defined by Eq. (32)
NA NA,av r Re Re ReN s sav Sc Sc Sh T Usg Usl v
Notation a A A0 A∞ Bo cA cA,i cA0 c∗ C Ci D Dc Di DT F Fr g G h H¯ JA kL K KL,av KL a l n N
specific interfacial surface, m2 /m3 exposed area of time-dependent surface, m2 initial area of the surface exposed to gas, m2 total area of the surface exposed to gas, m2 Bond number defined by Eq. (79) time-dependent molar concentration of species A, mol/m3 molar concentration of species A at the interface, mol/L3 molar concentration of species A at the bulk of fluid, mol/m3 saturation concentration, mol/m3 dimensionless concentration profile defined by Eq. (27) constant molecular diffusivity, m2 /s column diameter, m impeller diameter, m tank diameter, m defined by Eq. (34) Froude number defined by Eq. (79) gravitational acceleration, m/s2 defined by Eq. (35) defined by Eq. (44) average distance of approach of eddies to interface, m molar flux of diffusion of species A relative to the molar average velocity, mol/m2 s fluid-film mass transfer coefficient, m/s consistency index in a power-law model, kg sn−2 /m time-average overall mass transfer coefficient, m/s overall volumetric mass transfer coefficient, 1/s length scale of eddies in viscous dissipation range, m dimensionless exposed surface area defined in Eq. (54); flow index in a power-law model impeller speed, rps
Greek letters ∇ ˙ c 0 ϑ
shear rate, 1/s vector differential operator defined by Eq. (9) unit vector, m volume fraction, dimensionless energy dissipation rate per unit mass, W/kg time, s circulation time around vessel, s characteristic constant defined for each system; e.g. for gas–fluid contactors 0 might be bubble formation time, s amplitude of period, 1/s kinematic viscosity, m2 sn−2 viscosity, kg/m s density, kg/m3 interfacial tension, N/m dimensionless variable used in Eq. (54) dimensionless variable defined by Eq. (35) surface–age distribution function, m2 /s
Subscripts c BM d FM g p
continuous phase bubble mean (diameter) dispersed phase fluid particle mean (diameter) gas particles in general
B. Jajuee et al. / Chemical Engineering Science 61 (2006) 3917 – 3929
Acknowledgments A. Margaritis, D. Karamanev, and M.A. Bergougnou acknowledge financial support from Imperial Oil Ltd., Sarnia, Ontario, and Natural Sciences and Engineering Research Council (NSERC) of Canada. B. Jajuee acknowledges support by the Western Engineering Scholarships. References Akita, K., Yoshida, F., 1973. Gas holdup and volumetric mass transfer coefficient in bubble columns. Industrial & Engineering Chemistry Process Design and Development 12, 76–80. Angelo, J.B., Lightfoot, E.N., 1968. Mass transfer across mobile interfaces. A.I.Ch.E. Journal 14 (4), 531–540. Angelo, J.B., Lightfoot, E.N., Howard, D.W., 1966. Generalization of the penetration theory for surface stretch: application to forming and oscillation drops. A.I.Ch.E. Journal 12 (4), 751–760. Banerjee, S., Scott, D., Rhodes, E., 1968. Mass transfer to falling wavy liquid films in turbulent flow. Industrial & Engineering Chemistry Fundamentals 7 (1), 22–27. Calderbank, P.H., 1958. Physical rate processes in industrial fermentation, Part I: the interfacial area in gas–liquid contacting with mechanical agitation. Transactions of the Institution of Chemical Engineers 36, 443–463. Dankwerts, P.V., 1951. Significance of liquid-film coefficients in gas absorption. Industrial & Engineering Chemistry 43, 1460–1467. Deckwer, W.-D., Nguyen-Tien, K., Schumpe, A., Serpemen, Y., 1982. Oxygen mass transfer into aerated CMC solutions in a bubble column. Biotechnology and Bioengineering 24, 461–481. Dobbins, W.E., 1956. The Nature of the Oxygen Transfer Coefficient in Aeration Systems. Biological Treatment of Sewage and Industrial Wastes, vol. 1, pt. 2-1. Reinhold, New York, pp. 141–148. Durst, F., Tsiklauri, G.V., Afgan, N.H., 1979. Two-Phase Momentum, Heat and Mass Transfer in Chemical, Process, and Energy Engineering Systems, vol. 2, Washington, pp. 835–876 El. Temtamy, S.A., Khalil, S.A., Nour-El-Din, A.A., Gaber, A., 1984. Oxygen mass transfer in a bubble column bioreactor containing lysed yeast suspensions. Applied Microbiology and Biotechnology 19, 376–381. Godbole, S.P., Schumpe, A., Shah, Y.T., Carr, N.L., 1984. Hydrodynamics and mass transfer in non-Newtonian solutions in a bubble column. A.I.Ch.E. Journal 30 (2), 213–220. Harriott, P., 1962. A random eddy modification of the penetration theory. Chemical Engineering Science 17 (3), 149–154.
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