Mass transfer in a membrane tube with turbulent flow of Newtonian and non-Newtonian fluids

Journal of Membrane Science 148 (1998) 45±57

Mass transfer in a membrane tube with turbulent ¯ow of Newtonian and non-Newtonian ¯uids Madan G. Parvatiyar* Department of Chemical Engineering, University of Cincinnati, Cincinnati, OH 45221, USA Received 30 December 1997; received in revised form 13 April 1998; accepted 28 May 1998

Abstract Spectral theory of turbulence has been applied to obtain expressions for the mass transfer coef®cient in the case of Newtonian and non-Newtonian ¯uids. Expression of mass transfer coef®cient obtained from the spectral theory of turbulence is then compared with that obtained from other hydrodynamic models. Similarities and advantages of these models are discussed. Mass transfer models developed on the basis of hydrodynamic consideration, show their potential in providing mechanistic aspect of transport behavior with respect to mixing and viscous stresses of the ¯uid on the membrane surface. The effects of ¯ow index and consistency factor of the non-Newtonian ¯uids, on the mass transfer phenomena in a membrane tube are discussed. # 1998 Published by Elsevier Science B.V. All rights reserved. Keywords: Mass transfer; Spectral theory; Ultra®ltration; Non-Newtonian ¯uids; Concentration polarization

1. Introduction Membranes play an important role in the science of industrial and biochemical processes. Its wide application, has led to the development of many theories describing the mechanism of separation during the transport process. One of the major concern, as to the membrane process, is the realization of the concentration polarization of the solute on the membrane surface. Many theoretical and experimental efforts have been made in the past to study these effects [1±18]. None of the older theories describe transport phenomena based on the energetics of the turbulent structure of the ¯uid ¯ow ®eld. In most of the time emphasis *Corresponding author. Tel.: +1-513-556-7378; fax: +1-513569-7105.

was given to the eddies and their diffusion characteristics. In the past, proper selection of the energy channel out of various energy transfer channels present in the ¯ow ®eld, was never addressed in order to estimate driving force for the mass transfer processes in a quantitative way. In this paper, attempt is made to develop expression for the mass transfer coef®cient solely based on the turbulent structure of the ¯uid ¯ow. For the purpose of mathematical development, a simple expression of the energy spectrum function was selected in the calculation of mass transfer coef®cient. A part of the total energy spectrum function consisting of all the small eddies compared on a Kolmogorov length scale, was considered for the quantitative evaluation of energy dissipation against the viscous forces. There is no current prediction of the energy spectrum in the viscous range that is strong

0376-7388/98/$ ± see front matter # 1998 Published by Elsevier Science B.V. All rights reserved. PII: S0376-7388(98)00146-X

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M.G. Parvatiyar / Journal of Membrane Science 148 (1998) 45±57

enough to explain the exact nature of the energy transfer between eddies. The ®rst formulation was due to Heisenberg [19], who argued that the effect of the smaller eddies i.e., components with wave numbers more than k, on the large ones is equivalent to an eddy viscosity. His form is S…k† ˆ

32 ÿ3=2 C 9 1

Zk

E…k1 †k12 dk1

0

Z1  k

E…k1 † k13

1=2

dk1

(1)

where C1 is the spectrum constant. The second one, due to Obukhov [20], assumes that the smaller eddies provide a Reynolds stress for the larger ones to work on, and that the Reynolds stress is proportional to the intensity of the smaller eddies. He obtained the expression: 2k 31=2 1 Z ÿ3=2 Z 8C1 4 2 5 E…k1 †k1 dk1 E…k1 †dk1 S…k† ˆ p 3 3 0

(2)

k

The third one, due to Kovasznay [21], is the simplest expression and it assumes the transfer to be determined by the spectral intensity E(k). Then, ÿ3=2

S…k† ˆ 2C1

‰E…k†Š3=2 k5=2

(3)

Within the inertial subrange, each form of the expression leads to the standard result E…k† ˆ C1 2=3 kÿ5=3

(4)

Another development due to Townsend [22], which closely resembles the ¯ow phenomena very close to the membrane tube involves the following descriptions: (1) supposes that the diffusive action of turbulence extends vortex ®laments and concentrates vorticity into sheets, thickness of which is governed by a balance between outward viscous diffusion and the lateral compression associated with the stretching, (2) assumes that the turbulent rates of the strain provide the lateral compression and resemble locally plane straining. Townsend showed that the spectrum function in the description range is

2. Spectral theory The viscous dissipation of energy in a turbulent ¯ow is generally written as Z1  ˆ 2

dkk2 E…k†

(6)

0

The energy dissipation rate can be further written from the Eqs. (5) and (6) as:  ˆ 3927:29 3 =l4d

(7)

where the eddy length is de®ned as ld ˆ 2=ks

(8)

and ks is the wave number characteristic of the dissipative eddies. The energy dissipation in a turbulent ¯ow is also related to the root mean square of the velocity ¯uctuation as follows:  ˆ …V1 †3 =ld

(9)

Assuming ¯uctuating velocity V1 equal to the frictional velocity V0, the frictional velocity can be written from Eqs. (7) and (9) as V0 ˆ 2:0…†1=4

(10)

The ratio of the thickness of the diffusion sublayer to that of viscous sublayer in the case of Newtonian ¯uid is discussed by Davies [23], and is expressed as 2 ˆ 1:8‰D=Š1=3 1

(11)

where viscous sublayer is de®ned as 1 ˆ 5=V0

(12)

Substituting the expression for the diffusion sublayer 2, in the mass transfer expression [23]: kL ˆ 0:67D=2

(13)

(5)

the expression for the mass transfer coef®cient becomes  2=3 D V0 (14) kL ˆ 0:075 

Using the above form of the energy spectra, correlations for the mass transfer are developed.

Thus, the mass transfer coef®cient in terms of energy dissipation rate is obtained from the Eqs. (10) and (14)

E…k† ˆ …8=†1=2 1=4  5=4 …k=ks †ÿ2 exp‰ÿ2…k=ks †2 Š

M.G. Parvatiyar / Journal of Membrane Science 148 (1998) 45±57

as kL ˆ 0:15Scÿ2=3 …†1=4

(15)

Mass transfer model developed on the spectral theory of turbulence, is then compared with that obtained from the hydrodynamic model. In the hydrodynamic model, while developing mass transfer coef®cient model, the velocity pro®les in the turbulent boundary layer are considered on an empirical way. One advantage of this hydrodynamic approach over the spectral theory is that, it provides a mechanistic picture of the transport process in terms of viscous and mixing stresses present in the ¯uid. This picture has draw back in the sense that the foundation is not based on the energy concept of the turbulent ®eld. Spectral theory is more thermodynamical in nature and follow the energy conservation principle. Mass transfer results obtained from these two different models are compared with each other and the expressions are obtained for the Sherwood number as a function of the generalized Reynolds number. These expressions for mass transfer coef®cient and concentration polarization are equally valid for the Newtonian as well as non-Newtonian ¯uids. In the hydrodynamic approach, mass transfer coef®cients are derived under three different approximation schemes viz. (1) Buffer layer approximation, (2) Turbulent core layer approximation, and (3) Viscous sublayer approximation. Expressions derived under these approximations, provide an exclusive picture of the mechanism underlying the mass transfer process. In the buffer layer approximation, it describes the mass transfer based on the concept of equal importance of viscous and mixing stresses. In the turbulent core approximation, the expression for mass transfer is based on the idea of mixing stress exclusively and rendering the viscous stress negligible. In the viscous sublayer approximation, mass transfer expression incorporates to some degree the in¯uence of viscous stress in the main mixing stress component of the ¯ow ®eld. 3. Hydrodynamic theory In the hydrodynamic approach, the dependence of mass transfer coef®cient is written as a function of position in the ¯ow ®eld. A generalized equation of the velocity pro®le of Doge±Metzner [24] together with

47

that of Clapp [25] have been employed to obtain mass transfer expressions under three different approximation schemes. Under these three approximation schemes, transport of mass in a membrane tube is then analyzed. Dodge and Metzner have developed an expression for the generalized turbulent velocity pro®le in the case of turbulent boundary layer and this is applicable to both Newtonian and non-Newtonian ¯uids. The equation in terms of nondimensional velocity u‡ and distance y‡, has the form u‡ ˆ

2:457 0:4 2:45Q ln y‡ ÿ 1:2 ‡ 0:75 n0:75 n n

(16)

where

  1 Q ˆ 1:96 ‡ 1:255n ÿ 0:7068n ln 3 ‡ n

(17)

On the other hand, Clapp proposed generalized velocity pro®les for the three separate layers of the boundary layer. The velocity pro®le in side the turbulent boundary layer according to Levitz three zone model is shown in Fig. 1(a). The velocity pro®les for the separate zones are given as: Viscous sublayer (0
(18)

Buffer layer (5n<‡): u‡ ˆ

5:0 ln y‡ ÿ 3:05 n

(19)

and the Turbulent core layer (y‡>‡): u‡ ˆ

2:78 3:8 ln y‡ ‡ n n

(20)

where u‡ ˆ

u V0

(21)

V02ÿn yn …K=†

(22)

and y‡ ˆ

The root-mean square values of the ¯uctuating velocities of the ¯ow ®eld are given by Laufer [26]: V1 ˆ V0 P0:5

(23)

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M.G. Parvatiyar / Journal of Membrane Science 148 (1998) 45±57

Fig. 1. (a) Levitz three zone model of the turbulent boundary layer. (b) Coordinate system in the membrane tube.

From Eqs. (26) and (27), we get     exp…RB † 1=2…n‡1† n=2…n‡1† K 1=2…n‡1† V0 ˆ  P1:5n 

where P is polynomial in X: 2

P ˆ 4:57 ÿ 1:94X ‡ 58:11X ÿ 554:69X

3

‡ 1871:3X 4 ÿ 3259:4X 5 ‡ 3192:33X 6 ÿ 1675X 7 ‡ 366:134X 8

(24)

and X ˆ1ÿ

r R

(25)

The coordinate `r' extends from the surface of the tube to the tube center line (R), Fig. 1(b) The normalized distance y‡ is now obtained from the Eqs. (9), (22) and (23) as: y‡ ˆ

2…n‡1†

V0 P1:5n n  …K=†

(26)

Here it has been assumed that the layer thickness is characterized by the eddy length scale. 3.1. Buffer layer approximation On equating the generalized velocity pro®le Eq. (16), with the buffer layer velocity pro®le Eq. (19), we get y‡ ˆ exp…RB †

(27)

3:05 ÿ …0:4=n1:2 † ‡ …2:45Q=n0:75 † …5=n† ÿ …2:457=n0:75 †

For the Newtonian case (nˆ1) and at Xˆ1, the above equation reduces to V0 ˆ 2:2…†0:25

(30)

Kawase and Ulbrecht [27] obtained expression for the mass transfer coef®cient in terms of frictional velocity, ¯ow index, and consistency factor of the ¯uid. The expression for the mass transfer coef®cient is given as:  ÿ2=3n …4ÿn†=3n 1=3 2=3 K V0 (31) kL ˆ 0:075n D  This is the generalized mass transfer coef®cient for the Newtonian and non-Newtonian ¯uids. Mass transfer coef®cient under buffer layer approximation is then obtained from Eqs. (29) and (31):  4ÿn=6n…n‡1† 1=3 2=3 exp…RB † kLB ˆ 0:075n D P1:5n  ÿ5=6…n‡1† K 4ÿn=6…n‡1† (32)  For the Newtonian case (nˆ1) and at Xˆ1, the above equation reduces to

where RB ˆ

(29)

(28)

kLB ˆ 0:166Scÿ2=3 …†1=4

(33)

M.G. Parvatiyar / Journal of Membrane Science 148 (1998) 45±57

3.2. Turbulent core approximation The nondimensionalized distance y‡, under the turbulent core approximation scheme is obtained from the Eqs. (16) and (20): y‡ ˆ exp…RT †

(34)

where RT ˆ

0:75

1:2

(35)

Frictional velocity under turbulent core approximation, can be now obtained from Eqs. (26) and (34) as:     exp…RT † 1=2…n‡1† n=2…n‡1† K 1=2…n‡1†  (36) V0 ˆ P1:5n  For the Newtonian ¯uid (nˆ1) and at the interface (Xˆ1), this reduces to V0 ˆ 2:68…†0:25

(37)

Mass transfer expression under turbulent core approximation is now obtained from the Eqs. (31) and (36).  4ÿn=6n…n‡1† 1=3 2=3 exp…RT † kLT ˆ 0:075n D P1:5n  ÿ5=6…n‡1† K 4ÿn=6…n‡1† (38)  For the Newtonian ¯uid (nˆ1) and Xˆ1, this has the expression ÿ2=3

…†

1=4

(39)

3.3. Viscous sublayer approximation The nondimensional distance under viscous sublayer approximation is obtained from Eqs. (16) and (18): y‡ ˆ exp…RA †

(40)

where RA ˆ 1:37n ‡ 1:71

(41)

The frictional velocity with viscous sublayer approximation is obtained from the Eqs. (26) and (40), the

(42)

For the Newtonian ¯uids (nˆ1) and at the interface (Xˆ1), this reduces to V0 ˆ 2:15…†0:25

…2:45Q=n † ÿ …3:8=n† ÿ …0:4=n † …2:78=n† ÿ …2:457=n0:75 †

kLT ˆ 0:20Sc

result is:     exp…RA † 1=2…n‡1† n=2…n‡1† K 1=2…n‡1† V0 ˆ  P1:5n 

49

(43)

From Eqs. (31) and (42), the mass transfer coef®cient with viscous sublayer approximation is written as   exp…RA † 4ÿn=6n…n‡1† kLA ˆ 0:075n1=3 D2=3 P1:5n  ÿ5=6…n‡1† K 4ÿn=6…n‡1† (44)  For the Newtonian ¯uids (nˆ1) and for Xˆ1, this simpli®es to kLA ˆ 0:16Scÿ2=3 …†1=4

(45)

The energy dissipation rate per unit mass of the ¯uid due to viscous dissipation in a membrane tube can be written as  ˆ 0:0314U 3…2ÿn† d 3nÿ4 ‰Regen Šÿ3=8

(46)

where the generalized Reynolds number is de®ned as  n U …2ÿn† dn 4n (47) Regen ˆ nÿ1 8 …K=† 3n ‡ 1 and the generalized Schmidt number as   1ÿn K d Sc ˆ  DU 1ÿn

(48)

The Sherwood number under buffer layer, turbulent core layer, and viscous sublayer schemes are obtained from Eqs. (32), (38) and (44), respectively and from the de®nition of the Sherwood number as: Sh ˆ

kL d D

(49)

The results obtained under the approximation schemes are: Buffer layer approximation (BL): ShBL ˆ ABL …n†…Sc †1=3 …Regen †116ÿ53n=96…2ÿn†

(50)

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M.G. Parvatiyar / Journal of Membrane Science 148 (1998) 45±57

Turbulent layer approximation (TL): ShTL ˆ ATL …n†…Sc †1=3 …Regen †116ÿ53n=96…2ÿn†

(51)

Sublayer approximation (AL): ShAL ˆ AAL …n†…Sc †1=3 …Regen †116ÿ53n=96…2ÿn†

(52)

where ABL, ATL, and AAL are given as ABL ˆ A…n†‰exp…RBL †Š4ÿn=3n…nÿ2†

(53)

4ÿn=3n…nÿ2†

(54)

4ÿn=3n…nÿ2†

(55)

ATL ˆ A…n†‰exp…RTL †Š

AAL ˆ A…n†‰exp…RAL †Š

The ¯ow index dependent coef®cient A(n) is de®ned as A…n† ˆ 0:075n1=3 …3927:3†4ÿn=12…nÿ2†  16nÿ7n2 =12…nÿ2† 4n  3n ‡ 1  …8†…1ÿn†…16ÿ7n†=12…nÿ2†  …0:0314†4ÿn=12…2ÿn†

(56)

The mathematical developments made so far are now applied to analyze the mass transfer behavior in the case of pectin solutions with different concentrations. The rheological properties [28] of the pectin solutions are given in Table 1. 4. Results and discussion The Schmidt number for a Newtonian ¯uid is de®ned as the ratio of kinematic viscosity to the diffusivity. For non-Newtonian ¯uids, the Schmidt Table 1 Rheological properties of pectin solutions Concentration of pectin solution (wt. %)

Flow index (n)

Consistency factor (K) (Pa sn)

0.355 0.41 0.537 0.606 0.62 0.84 0.93

1.0 0.9857 0.957 0.943 0.9397 0.9 0.8857

0.0024 0.00408 0.0066 0.008357 0.00885 0.0212 0.0296

number additionally depends on the rheological parameters of the solution and its ¯ow velocity. The Schmidt number decreases with the increase in ¯ow velocity of the ¯uid. The higher the concentration of the solution, higher the in¯uence of ¯ow velocity is on the Schmidt number. Results of the Schmidt number as a function of the generalized Reynolds number, in the case of pectin solutions with concentrations 0.93(nˆ0.8857, Kˆ0.0296 Pa sn), 0.84(nˆ0.9, Kˆ0.0212 Pa sn), 0.606(nˆ0.928, Kˆ0.00835 Pa sn), 0.537(nˆ0.957, Kˆ0.0066 Pa sn), and 0.41(nˆ0.9857, Kˆ0.00408 Pa sn) wt. % are shown in Fig. 2. The effect of ¯ow velocity on the Schmidt number is lower for the pectin solutions with 0.41, 0.537, and 0.606 wt. % concentrations. Furthermore, the pectin solution with 0.606 wt. % concentration has lower values of the Schmidt number compared to the pectin solutions with 0.537 and 0.41 wt. % concentrations. This nonlinear variation of the Schmidt number with the generalized Reynolds number can be attributed to the rheological properties of the solutions. The role of the consistency factor (K) is analogous to the viscosity of the Newtonian ¯uid, where as the ¯ow index (n) of the solution is indicative of the dependence of stress on the shear rate. The dependence of the mass transfer coef®cient or the Sherwood number on the generalized Reynolds number under three different approximation schemes are shown in Fig. 3(a±c). In the buffer layer approximation where the viscous and mixing stresses are the dominant mechanisms for the mass transfer processes, the Sherwood number for 0.93 wt. % of pectin solution (nˆ0.8857, Kˆ0.029 Pa sn) shows a high dependence on the generalized Reynolds number, Fig. 3(a). Among the solutions with concentration 0.84, 0.6, and 0.41 wt. %, respectively, the effect of generalized Reynolds number on the Sherwood number is lower than compared to that obtained from the 0.93 wt. % solution. The effect of the Reynolds number in the case of solution (Newtonian ¯uid) with 0.355 wt. % concentration (nˆ1, Kˆ0.0024 Pa sn), on the Sherwood number is the lowest compared to the values obtained in the cases of the non-Newtonian solutions. Predictions of the Sherwood number as a function of the generalized Reynolds number under turbulent core approximation scheme are shown in Fig. 3(b). The results for the Sherwood number are higher compared to that obtained from the buffer layer

M.G. Parvatiyar / Journal of Membrane Science 148 (1998) 45±57

51

Fig. 2. Variation of Schmidt number (Sc*) with Reynolds number (Regen).

approximation schemes. Solution with 0.923 wt. % concentration (nˆ0.8857, Kˆ0.0296 Pa sn), shows higher in¯uence of the generalized Reynolds number on the Sherwood number. For solutions having concentrations 0.84(nˆ0.9, Kˆ0.0212 Pa sn), 0.606(nˆ 0.943, Kˆ0.008357 Pa sn), and 0.41(nˆ0.9857, Kˆ 0.00408 Pa sn) wt. %, respectively, the in¯uence of the generalized Reynolds number on the Sherwood number is lower than that obtained from 0.923 wt. % solution. The value of the Sherwood number decreases with the decrease in pectin concentration in the solution. The in¯uence of the Reynolds number on the Sherwood number is the lowest for solution (Newtonian ¯uid) with 0.355(nˆ1.0, Kˆ0.0024 Pa sn) wt. % concentration. The results of the Sherwood number as a function of the generalized Reynolds number under viscous sublayer approximation scheme are shown in Fig. 3(c). Under sublayer approximation scheme pectin solution with 0.9287 wt. % concentration again shows a high sensitivity of the Sherwood number with the generalized Reynolds number. As the concentration of the pectin solution decreases from 0.9286 wt. % to 0.41 wt. %, there becomes a drastic change in sensitivity. The reduction of sensitivity on the Sherwood number with the generalized Reynolds number for pectin solutions with 0.41, 0.84, and 0.606 wt. % concentrations, is gradual. However,

the values of the Sherwood number as a function of the Reynolds number, in the case of Newtonian ¯uid with 0.335 wt. % concentration (nˆ1.0, Kˆ 0.0024 Pa sn), are much lower than the values obtained in the case of the non-Newtonian solutions. Comparison of the Sherwood number predicted as a function of generalized Reynolds number under different approximation schemes are shown in Fig. 4(a±g). Solutions with 0.9287 and 0.84 wt. % concentrations show higher values of the Sherwood number as a function of the generalized Reynolds number under turbulent core approximation schemes than compared to the values obtained from buffer layer or viscous sublayer approximation schemes, Fig. 4(a± b). The values of the Sherwood number under sublayer approximation are lower than that obtained from buffer layer approximation scheme. The Sherwood numbers for solutions with concentration of 0.62(nˆ0.939, Kˆ0.00885 Pa sn), 0.606(nˆ0.943, Kˆ0.008357 Pa sn), and 0.537(nˆ0.957, Kˆ 0.006 Pa sn) wt. %, have very close values to that of buffer layer and sublayer schemes. However, the results for the Sherwood number obtained under similar conditions and with the turbulent core approximation scheme are lower than that obtained from the buffer layer and the sublayer schemes, Fig. 4(c±e). Solutions with concentration 0.41(nˆ0.9857,

52

M.G. Parvatiyar / Journal of Membrane Science 148 (1998) 45±57

Fig. 3. (a) Variation of Sherwood number (ShBL) with Reynolds number (Regen). (b) Variation of Sherwood number (ShTL) with Reynolds number (Regen). (c) Variation of Sherwood number (ShAL) with Reynolds number (Regen).

M.G. Parvatiyar / Journal of Membrane Science 148 (1998) 45±57

Kˆ0.00408 Pa sn) and 0.355(nˆ1, Kˆ0.024 Pa sn) wt. %, respectively, show a different trend for the Sherwood number as a function of the generalized Reynolds number. In both the cases, the predictions of the Sherwood number as a function of the generalized Reynolds number and under turbulent core layer

53

approximation, are much lower compared to that obtained from the buffer layer and the sublayer approximation schemes Fig. 4(f±g). These nonlinear behavior of the non-Newtonian ¯uids can be attributed to the rheological properties of the solution. It is indicative that the ¯ow index and the consistency

Fig. 4. (a) Variation of Sherwood number with Reynolds number under different approximation schemes (nˆ0.8859, Kˆ0.02957 Pa sn). (b) Variation of Sherwood number with Reynolds number under different approximation schemes (nˆ0.9, Kˆ0.0212 Pa sn). (c) Variation of Sherwood number with Reynolds number under different approximation schemes (nˆ0.9397, Kˆ0.00885 Pa sn). (d) Variation of Sherwood number with Reynolds number under different approximation schemes (nˆ0.9428, Kˆ0.00835 Pa sn). (e) Variation of Sherwood number with Reynolds number under different approximation schemes (nˆ0.957, Kˆ0.006604 Pa sn). (f) Variation of Sherwood number with Reynolds number under different approximation schemes (nˆ0.9857, Kˆ0.00408 Pa sn). (g) Variation of Sherwood number with Reynolds number under different approximation schemes (nˆ1.0, Kˆ0.0024 Pa sn).

54

M.G. Parvatiyar / Journal of Membrane Science 148 (1998) 45±57

Fig. 4. (Continued)

M.G. Parvatiyar / Journal of Membrane Science 148 (1998) 45±57

55

Fig. 4. (Continued)

factor of the solution have direct in¯uence on the viscous and mixing stresses of the ¯uid. It may be the fact that the role of turbulent mixing in mass transfer is impeded by the increased non-Newtonian character of the ¯uid where as the viscous stresses are less affected. 5. Conclusion Theoretical models for describing the mass transfer phenomena in a membrane tube and in the case of Newtonian and non-Newtonian ¯uids, are developed.

From the consideration of the energy spectrum of the ¯ow ®eld, the expressions for mass transfer coef®cient are obtained. Mass transfer models are also obtained from the hydrodynamic consideration of the ¯ow ®eld. Similarities and advantages of these models are discussed. The in¯uence of the rheological parameters of the ¯uid on mass transfer has been emphasized. It is also concluded that the viscous stresses are more sensitive to the rheological parameters of the solution during mass transfer processes compared to the mixing stresses present in turbulent layer and buffer layer approximation schemes.

56

M.G. Parvatiyar / Journal of Membrane Science 148 (1998) 45±57

6. List of symbols AAL ABL ATL C1 D d E(k) J K k kL  kLB  kLT  kLA ks ld n P RA RB RT Re Regen Rm r S(k) Sc* Sh* U u‡ V0 V1 vp X y‡

Defined in Eq. (55) Defined in Eq. (53) defined in Eq. (55) Spectrum constant Diffusivity Membrane tube diameter Energy spectrum, Eq. (4) Permeate flux Consistency factor Wave number Mass transfer coefficient, Eq. (13) Mass transfer coefficient, Eq. (32) Mass transfer coefficient, Eq. (38) Mass transfer coefficient, Eq. (44) Wave number characteristic to dissipative eddies Eddy length scale, Eq. (8) Fluid flow index Polynomial in X, Eq. (24) Defined in Eq. (41) Defined in Eq. (28) Defined in Eq. (35) Reynolds number Defined in Eq. (47) Membrane resistance Coordinate from the surface of the tube to the center of the tube (R) Energy spectra Schmidt number, Eq. (48) Sherwood number Velocity of the fluid Nondimensional velocity Frictional velocity Fluctuating velocity Permeation velocity Coordinate, defined in Eq. (25) Nondimensional distance

Subscript BL TL AL

Buffer layer approximation Turbulent core layer approximation Viscous sublayer approximation

Greek letters 1 2    

Viscous sublayer thickness Diffusion sublayer thickness Energy dissipation rate per unit mass Kinematic viscosity Viscosity Density of the fluid

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