Performance prediction of membrane modules incorporating the effects of suction in the mass transfer coefficient under turbulent flow conditions

Separation and Purification Technology 55 (2007) 182–190

Performance prediction of membrane modules incorporating the effects of suction in the mass transfer coefficient under turbulent flow conditions Saurabh Bhatia, Sunando DasGupta, Sirshendu De ∗ Department of Chemical Engineering, Indian Institute of Technology, Kharagpur, Kharagpur 721302, India Received 6 September 2006; received in revised form 27 November 2006; accepted 27 November 2006

Abstract Effects of suction are included in the mass transfer coefficient to predict the performance of spiral wound and tubular modules under the turbulent flow conditions. The analysis is valid for the osmotic pressure controlled filtration. Dextran is selected as the model solute. Variations of bulk concentration, bulk velocity, retentate channel pressure, permeate flux, permeate concentration, etc., along the module length are obtained after solving the design equations numerically. The results are compared with the standard module design using mass transfer coefficient without including suction and the film theory. The results indicate that without including suction in the mass transfer coefficient leads to gross underprediction of the length averaged permeate flux (about 70–75%) and overprediction of the average permeate of concentration (about 8–10%) for various operating conditions studied herein (for a 10 m long module). On the other hand, calculations using film theory leads to underprediction of average permeate flux in the range of 30–35%. © 2006 Elsevier B.V. All rights reserved. Keywords: Turbulent flow; Suction; Permeate flux; Permeate concentration; Film theory

1. Introduction Membrane modules, namely, spiral wound, tubular, hollow fiber, etc., are the final equipment to be installed in order to run a membrane based plant. A prior knowledge of the performance of the modules helps in two ways; first, it leads to a proper design (i.e., module length, channel height, tube diameter, etc.) to meet the required output, i.e., average permeate flux and permeate quality. Secondly, given a designed module, the output values can be obtained at various operating conditions without incurring any cost to conduct the trial experimental runs. The phenomenon, concentration polarization, i.e., the deposition of the solute particles over the membrane surface makes the design of the membrane modules complex [1–8]. The simplest and the earliest model for concentration polarization is film theory [1]. In depth discussions of the shortcoming of film theory are available in literature [9–11]. The major drawbacks include: (i) constant thickness of the mass transfer boundary layer, (ii) the mass transfer coefficient is obtained from the heat and mass transfer analogy for the non-porous conduits. There-

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fore, calculations using film theory results in underprediction of the permeate flux for the osmotic pressure controlled membrane filtration. Detailed and more accurate models in two dimensions are generated using computational fluid dynamics for flow in complex geometries. A variety of techniques were employed for solution. Huang and Morrissey [7] used finite element method, Lee and Clark [12] used finite difference technique. Hansen et al. [13] used a spectral method for simultaneous solution of Navier Stokes equations coupled with convective-diffusive solute balance equations. Geraldes et al. [14] and Wiley and Fletcher [15] used finite volume methods for simulations. The predictions from these models are accurate but highly computation intensive. Moreover, most of these models are developed for laminar flow regime. Therefore, simplistic but efficient one dimensional models with reasonable accuracy for turbulent flow domain are still relevant and useful to the process engineers. A typical module design includes the development of overall material balance, component balance and energy balance equations in a differential element. Film theory along with Darcy’s law is used to relate the membrane surface concentration to the bulk concentration. Corresponding mass transfer coefficient is obtained from heat and mass transfer analogy for the non-porous conduits; for example, Dittus-Boelter equation for the turbulent

S. Bhatia et al. / Separation and Purification Technology 55 (2007) 182–190

Nomenclature A membrane area (m2 ) A1,2,3 coefficients in Eq. (19) B1 , B2 , B3 virial coefficients in Eq. (15) c bulk solute concentration (kg/m3 ) membrane surface concentration (kg/m3 ) cm cp permeate concentration (kg/m3 ) c0 inlet solute concentration (kg/m3 ) d tube diameter (m) de equivalent diameter (m) D solute diffusivity (m2 /s) f friction factor h channel half height (m) I definite integral in Eq. (21) k mass transfer coefficient (m/s) L channel length (m) Lp membrane permeability (m3 /(N s)) P guage pressure (Pa) P0 inlet guage pressure (Pa) P transmembrane pressure (Pa) Q volumetric flow rate (m3 /s) Re Reynolds number (ρude /μ) Rr intrinsic retention Sc Schmidt number (μ/ρD) u bulk velocity (m/s) u0 bulk velocity (m/s) vw local permeate flux (m3 /(m2 s)) vw length averaged permeate flux (m3 /(m2 s)) w channel width (m) x axial coordinate (m) y transverse coordinate (m) Greek letters λ parameter in Eq. (22) η dummy variable in Eq. (21) μ viscosity (Pa s) π osmotic pressure (Pa) πm osmotic pressure at the membrane surface (Pa) osmotic pressure of the permeate side (Pa) πp π osmotic pressure difference across membrane (Pa) ρ solution density (kg/m3 )

regime [16]. The approximate analytical solutions for permeate flux using these correlations for mass transfer coefficient are available [17–20]. However, they have several limitations which are discussed in detail [21,22]. The assumptions like, linear relation of osmotic pressure with concentration and the ratio of the permeate flux to the mass transfer coefficient less than 0.2 simplified the algebra to obtain the analytical solutions of the design equations [17]. It may be noted that the above assumptions may be valid for reverse osmosis but are untenable for more porous ultrafiltration (UF) membranes. Moreover, the effects of suction are not included in the Sherwood number relations, like, Dittus-

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Boelter. The effects of suction on mass transfer coefficient in both laminar and turbulent flow regimes are already quantified [23,24]. These effects are found to be quite important for porous ultrafiltration membranes [24]. To overcome the above limitations, the design equations for membrane modules are modified and solved including the effects of suction in mass transfer coefficient under laminar flow conditions [25]. In the present study, the design equations are developed and numerically solved for the turbulent flow regime which is the most common in practice. The module performance with and without including the effects of suction in the mass transfer coefficient is compared. It is stressed that the present paper deals with permeation through the membrane in all cases signifying that there will always be permeation (permeate flux) through the membrane. However, as suction through the membrane can significantly alter the values of the mass transfer coefficient, this study specifically aims for a quantification of such an effect on mass transfer coefficient. In all the relevant figures of this work, it is therefore clearly mentioned in the legend that whether suction effect on mass transfer coefficient is considered or not. A comparison using Film theory (without including the effects of suction in mass transfer coefficient) is also presented. Finally, the developed design equations are applied to predict the UF data available in the literature [26]. 2. Theory 2.1. Flow through a spiral wound module Flow through a spiral wound module can be modeled as flow through a rectangular channel, when the module is opened up. The schematic of the flow through the rectangular channel under turbulent flow conditions is shown in Fig. 1. In order to obtain the governing equations of the velocity, solute concentration and transmembrane pressure drop, a differential element of the channel of length x is considered as shown in Fig. 2. 2.1.1. Material balance equation A material balance of the solution in the differential element (assuming no variation of the solution density) results in the following equation, dQ = −2wvw dx

Fig. 1. Schematic of the geometry of the flow channel.

(1)

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simplified to the following equation, ρfu2 du wvw 2 wvw 2 dP = −ρu + ρu − ρvw − dx dx Q Q 2h

(8)

Replacing the flow rate, Q from Eq. (2) and du/dx from Eq. (3), the governing equation of the transmembrane pressure drop across the module length is obtained as,   dP ρ v3 (9) = 3uvw − w − fu2 dx 2h u The fanning friction factor, f for the turbulent flow in a smooth pipe is given by Blasius formula [27],

Fig. 2. Schematic of the differential element in the module.

The volumetric flow rate is related to the cross sectional average velocity as, Q = 2hwu

(2)

Combining Eqs. (1) and (2), the governing equation of the average velocity is obtained, vw du =− dx h

(3)

2.1.2. Component balance equation The solute mass balance in the differential element as shown in Fig. 2 results, vw cp d(uc) =− dx h

(4)

Using Eq. (3), the governing equation of the solute in the bulk is obtained as, vw dc = (c − cp ) u dx h

(5)

where, cp is the permeate concentration. 2.1.3. Energy balance equation The energy balance equation in the differential element becomes, P − (P + dP) + =

ρu2 2

4ρf dx u2 2de

−

ρ(u + du) Avw ρ 2 + (u − v2w ) 2 Q 2 2

2.1.4. Darcy’s law for flow through the porous membrane Solvent flow rate through the membrane is expressed by Darcy’s law, vw = Lp (P − π)

(11)

Osmotic pressure of any macromolecular solution is expressed as, π = B 1 c + B 2 c 2 + B3 c 3

(12)

Therefore, the osmotic pressure difference across the membrane is given as, π = πm − πp

(13)

The intrinsic retention Rr , which is constant for a membrane solute system [28], relates the permeate concentration to the membrane surface concentration as follows: cp Rr = 1 − (14) cm Using Eqs. (11)–(14), the permeate flux can be expressed as function of membrane surface concentration: 2 vw = Lp [P − B1 cm Rr − B2 cm {1 − (1 − Rr )2 }

(6)

where, A is the area of the permeation. In the differential element, A = 2w dx. It may be noted here that P in Eq. (6) is the gauge pressure and since the permeate is collected at atmospheric pressure, P in Eq. (6) is equivalent to P, the transmembrane pressure drop. Inserting the definition of A in Eq. (6), the following equation is obtained, −dP − ρ(u du) +

0.079 (10) Re0.25 In Eq. (10), Reynolds number is defined based on the equivalent diameter. f =

wvw 2 wvw 2 4ρfu2 dx ρu dx − ρvw dx = Q Q 2de (7)

The equivalent diameter is defined as de ≈ 4 h for a thin channel. Using the definition of equivalent diameter, Eq. (7) can be

3 − B3 cm {1 − (1 − Rr )3 }]

(15)

2.1.5. Relation between the bulk and membrane surface concentration and mass transfer coefficient Bulk and the membrane surface concentration at any xlocation is connected through the definition of mass transfer coefficient,   ∂c k(cm − c) = −D (16) ∂y y=0 At the steady state, sum of all fluxes at the membrane surface is zero. This leads to the following equation,   ∂c vw (cm − cp ) = −D (17) ∂y y=0

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Combining Eqs. (16) and (17), the following equation is obtained, k(cm − cp ) = vw cm Rr

(18)

Combining Eqs. (15)–(18), the following algebraic equation for the membrane surface concentration in terms of bulk concentration and mass transfer coefficient is obtained, k(cm − c) 2 3 = [P − (A1 cm + A2 cm + A3 cm )] cm Rr Lp where, A1 = B1 Rr ; A3 = B3 [1 − (1 − Rr )3 ].

A2 = B2 [1−(1 − Rr )2 ]

(19) and

2.1.6. Incorporation of suction in mass transfer coefficient The mass transfer coefficient for the turbulent flow can be expressed as function of x and u as [24], 1/3  0.236D Re1.75 Sc k(x) = (20) I xde2 where, the integral I is calculated as,  3   ∞ η I= exp − − 2.82λη dη 3 0

(21)

In the above equation, λ is the suction parameter and is defined as [24], λ=

vw 1/3

D[Re1.75 Sc/d2e L]

(22)

where, vw is the length averaged permeate flux defined as,  1 L vw = vw (x)dx (23) L 0 For impervious conduit, the value of λ is zero and the value of I becomes 1.29. The relevant equations for the calculation based on the film theory with the mass transfer coefficient without suction is presented in the appendix. 2.2. Flow through a tubular module For flow through a tubular module of diameter “d”, the material balance, component balance and energy balance equations over the differential element become, du 4vw =− dx d dc 4vw = (c − cp ) dx d   dP 2ρ v3w 2 = 3uvw − − fu dx d u

u

(24) (25) (26)

The mass transfer coefficient for the tubular module is derived by Minnikanti et al. [24] and Eqs. (20)–(23) remain in the same form; only de is replaced by the tube diameter d.

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3. Numerical scheme The differential equations (Eqs. (3), (5) and (9)) are coupled with the algebraic equation, Eq. (19). The initial conditions of Eqs. (3), (5) and (9) are at x = 0, u = u0 , c = c0 and P = P0 . These coupled differential equations are solved using fourth order Runge-Kutta method and at each step of the numerical integration, the algebraic equation (Eq. (19)) is solved using Newton-Raphson method to update the value of cm . At every step of integration, it is required to know the value of mass transfer coefficient, k. For no suction case (λ = 0), it can directly be determined using Eqs. (20) and (21). But for the case of suction through the membrane, k is dependent on the unknown parameter, the length averaged permeate flux, vw . Therefore, a value of vw is assumed first and Eqs. (3), (5), (9) and (19) are solved simultaneously to obtain the profile of the permeate flux. The length averaged permeate flux is then calculated using Simpson’s one-third rule and checked against the assumed value of vw . The entire calculation is repeated until two successive values are within a small tolerance (10−9 m3 /(m2 s)). The detailed algorithm of numerical scheme is presented in Fig. A1 in appendix. 4. Results and discussions The model solute considered in this study is dextran. The solution and solute properties, membrane permeability, system geometry, etc., are presented in Table 1. As discussed in Section 3, the governing equations, i.e, Eqs. (3), (5), (9) and (19) are solved simultaneously to obtain the profiles of bulk concentration, bulk velocity, transmembrane pressure drop, permeate flux, membrane surface concentration, etc., along the module length. Before discussing the effects of suction on various performance parameters of the module, the influence of suction on mass transfer coefficient is presented in Fig. 3, for typical operating conditions of dextran solution. It is observed from Fig. 3 that at the end of the channel, the mass transfer coefficient with suction is about three times to that without suction. This clearly indicates that the enhanced mass transfer coefficient with suction alters the distribution of hydrodynamic and other dependent parameters which strongly influences the performance of the module. Variation of the membrane surface concentration along the module length is shown in Fig. 4 for typical operating condiTable 1 Data for simulation Solute

Dextran

Membrane permeability (m3 /N s) Solute diffusivity (m2 /s) Solution viscosity (Pa s) Solution density (kg/m3 ) Osmotic pressure (Pa)

2.46 × 10−11 6.75 × 10−11 0.001 1000.0 37.5c + 0.752 c2 + 0.00764 c3 , c in (kg/m3 ) 7,125–11,875 10.0 0.0095

Range of Reynolds number Module length (m) Equivalent diameter of the channel (m)

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tions of dextran. The profile of cm shows the usual trend, i.e., cm increases rapidly near the entrance of the channel and slowly in the downstream. This observation is also reported in the literature where variation of membrane surface at different axial locations are presented (Fig. 3 of [12]). It is also clear from their results that the membrane surface concentration increases with axial distance and the increase becomes gradual in the downstream of the channel. The results presented in this paper exactly corroborate this observation. This is due to the concentration polarization effects which gradually stabilizes by the external forced convection. The effects of suction are also evident from this figure. It is observed from the figure that the dimensionless membrane surface concentration decreases from about 6.7 to about 6.2 when suction is incorporated in the mass transfer coefficient. This is consistent with the results presented in Fig. 3. Since mass transfer coefficient increases with suction, more solutes are transported back from the membrane surface to the bulk of the solution, resulting in a reduction of membrane surface concentration. It may also be observed that for the case of film theory, cm profile lies in between the curve with suction (curve 3) and without suction in the mass transfer coefficient (curve 1). In the film theory, it is assumed that the thickness of the mass transfer boundary layer is constant over

the differential element (on which the governing equations are formulated). On the other hand, this assumption is not made for solution of the case with incorporating the suction in the mass transfer coefficient (presented by curve 3) [20]. Curve 1 is a special case of curve 3 (by putting λ = 0 in Eq. (21)). Therefore, although the differential element is sufficiently small, but because of the assumption of constant thickness of the mass transfer boundary layer (over the differential element), the film theory without suction predicts slightly lower value of the membrane surface concentration compared to the present formulation without suction in the mass transfer coefficient. Hence, the profile of membrane surface concentration calculated by film theory without suction (curve 2) lies below to that calculated by present work without suction. At the end of the channel, for the operating conditions in Fig. 3, the value of cm using present study without suction is 6.7 and it is about 6.5 using film theory at the same location. Although, it seems that the cm profiles are quite close to each other but it leads to substantial difference in the length averaged permeate flux as discussed subsequently. Variation of the permeate flux along the module length for the same operating conditions (as in Fig. 4) is shown in Fig. 5. The figure shows the expected trend that the permeate flux decreases along the module length. The decrease of flux is quite sharp at the entrance and gradual at the downstream. As discussed earlier, the membrane surface concentration builds up rapidly and then slowly along the module length, resulting in an increase of the osmotic pressure at the solution–membrane interface. This leads to a reduction in the available driving force causing the decline in the permeate flux along the module. The effects of suction on the profile of permeate flux are also apparent from Fig. 5. As discussed earlier (refer Fig. 3), the mass transfer coefficient increases with suction significantly. This enhances the backward transport of solutes from the membrane surface to the bulk. At the steady state this backward flux must match the flux across the membrane. Therefore, the permeate flux profile with suction is always above to that of without suction (compare curves 3 and 1). As discussed earlier, in case of film theory without suction, the thickness of mass transfer boundary layer is assumed to be constant (over the differential element) which is contrary to the actually developing boundary layer. This pseudo steady (with

Fig. 4. Variation of the membrane surface concentration along the module length. The operating conditions are same as in Fig. 3.

Fig. 5. Variation of the permeate flux along the module length. The operating conditions are same as in Fig. 3.

Fig. 3. Variation of the mass transfer coefficient along the module length for P0 =550 kPa; u0 = 1 m/s (Re = 9500); c0 = 50 kg/m3 and Rr = 1.0.

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Fig. 6. Variation of the bulk concentration along the module length. The operating conditions are same as in Fig. 3.

Fig. 8. Variation of the axial transmembrane pressure drop along the module length. The operating conditions are same as in Fig. 3.

respect to length) approximation and the fact that the effects of suction are not included in the mass transfer coefficient in the formulation of film theory, lead to the underprediction of permeate flux (curve 2) compared to curve 3. At the end of the channel, the dimensionless permeate flux has the value of 0.22 for the present study with suction in the mass transfer coefficient. This value is 0.12 for film theory and 0.05 for the present study without suction. Variation of the bulk concentration along the module length is shown in Fig. 6. This figure shows the expected trend. As more solvent permeates out of the module, the bulk concentration of the solute increases in the retentate side. As discussed earlier, the permeate flux is more for the formulation incorporating the suction effects in mass transfer coefficient, the increase in bulk concentration is maximum for this case (curve 3) compared to no suction cases (curves 1 and 2). The profiles of bulk velocity and the axial transmembrane pressure drop along the module length are shown in Figs. 7 and 8, respectively. Fig. 7 shows that the bulk velocity decreases along the module length. This is quite obvious as more solvent is permeated, the bulk velocity decreases. This is more pronounced for the case of suction (curve 3) compared to no suction cases (curves 1 and 2) as in the case of suction, more solvent is extracted. It is again mentioned that no suction in k, does not

mean no permeation through the membrane. Since porous membranes are present on the walls of the conduit and solvent flows out, the bulk velocity (u) decreases along the module length as per mass conservation (as shown in Fig. 7). The profiles of axial pressure drop increases linearly along the module length. The profiles for with and without suction have been almost coinciding indicating that suction has insignificant effect on the axial pressure drop. Since the reduction in velocity (Fig. 7) in all cases is marginal, friction factor, f remains almost constant resulting in near equal values of axial pressure drop. It is interesting to note that the value of pressure drop at the end of the module (10 m) is about 17 kPa which is about only 3% of the inlet pressure of 550 kPa. Variations of the length averaged permeate flux with the inlet transmembrane pressure drop at various cross flow velocities are shown in Fig. 9, for various formulations. The figure shows the expected trend that the average permeate flux increases with the

Fig. 7. Variation of the bulk velocity along the module length. The operating conditions are same as in Fig. 3.

Fig. 9. Variation of length averaged permeate flux with the pressure difference with various cross flow velocities: u0 = 1 m/s (Re = 9500); c0 = 50 kg/m3 and Rr = 1.0; curves 1, 2, 3 are for present study with suction; curve 1: uo = 1.25 m/s (Re = 11,875); curve 2: uo = 1.0 m/s (Re = 9500); curve 3: uo = 0.75 m/s (Re = 7125); curves 4, 5, 6 are for film theory; curve 4: uo = 1.25 m/s (Re = 11,875); curve 5: uo = 1.0 m/s (Re = 9500); curve 6: uo = 0.75 m/s (Re = 7125); curves 7, 8, 9 are for present study without suction; curve 7: uo = 1.25 m/s (Re = 11,875); curve 8: uo = 1.0 m/s (Re = 9500); curve 9: uo = 0.75 m/s (Re = 7125).

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pressure drop. Increase is higher at lower pressure and gradual at higher pressure. At higher operating pressure, more solutes are accumulated over the membrane surface resulting in an increase in the osmotic pressure at the solution–membrane interface, thereby reducing the effective driving force for the transport of the solvent across the membrane leading to a reduction in the rate of increase of the permeate flux. The effects of incorporation of suction in the mass transfer coefficient on the average flux are also depicted in the figure. As described earlier, suction increases the rate of solute transport from the membrane surface to the bulk solution. Therefore, the value of cm is the minimum for the present study with suction and maximum for that without suction (refer Fig. 3). Consequently, the flux values at a fixed transmembrane operating pressure are maximum for the present study with suction (curves 1–3) minimum for that without suction (curve 4–6). The prediction of the flux using film theory is in between the two, represented by curves 3–5, for various cross flow velocities. At 400 kPa pressure and u0 = 1.25 m/s, the value of permeate flux is about 3 × 10−6 m3 /(m2 s) for the present study with suction and this value without suction at the same operating conditions is about 0.8 × 10−6 m3 /(m2 s) resulting about 73% underprediction. On the other hand, at the same conditions, calculation using film theory results the flux value as about 2.1 × 10−6 m3 /(m2 s) indicting about 30% underprediction. Variation of the length averaged permeate concentration with the inlet transmembrane pressure drop for real retention 0.95 is shown in Fig. 10. The figure shows that the average permeate concentration increases with pressure. This is consistent as the length averages membrane surface concentration increases with pressure due to concentration polarization, more solutes permeate through the membrane. Since the value of the membrane surface concentration is more compared to that without suction, the permeate concentration is maximum for the present study without suction among the three formulations presented in the figure. Variation of the ratio of the length averaged permeate flux using the present study with suction to other formulations is

Fig. 10. Variation of the length average permeate concentration with the inlet pressure drop at Rr = 0.95, u0 = 1 m/s (Re = 9500) and c0 = 50 kg/m3 . Curve 1 is for present study without suction, curve 2 is for film theory and curve 3 is for present study with suction.

Fig. 11. Variation of length averaged permeate flux ratio with the membrane length at u0 = 1 m/s (Re = 9500); c0 = 50 kg/m3 ; Rr = 1.0 and P0 = 550 kPa.

presented in Fig. 11 for as a function of the module length. The figure clearly shows that for the operating conditions in Fig. 10, the value of the average flux using present study with suction in mass transfer coefficient is about 3.4–3.2 times to that without suction for the module length up to 10 m. Whereas, the present study with suction (in mass transfer coefficient) results in about 1.7–1.4 times of the flux predicted by film theory. In fact, this ratio (R2 ) decreases with the module length. Therefore, it may be expected that for quite long membrane module, the predictive power of film theory will be closer to that of the present study with suction. This is because for longer modules, the mass transfer boundary layer will be fully developed in the downstream of the module and the assumption that the constant thickness of the boundary layer becomes quite valid for most of the module length. An interesting comparison is also presented in this figure. Dittus-Boelter correlation is widely used for estimation of mass transfer coefficient in case of the turbulent flow regime inside a conduit. Using this correlation, the present problem is solved by replacing k in the film theory formulation (presented in the appendix) by the corresponding mass transfer coefficient obtained from Dittus-Boelter correlation (presented in appendix). The length averaged permeate flux using this correlation for mass transfer coefficient and its comparison with the present study with suction is also included in Fig. 11. It is observed from the figure that the ratio (R3 ) is about 0.7 for 1 m of the module length and this ratio decreases further for longer modules. Therefore, use of Dittus-Boelter correlation results in an overprediction of the average permeate flux and the extent of overprediction increases with the module length. This point is elaborated by comparing of the prediction using different methods with the experimental data, taken from the work of Wijmans et al. [26]. In their work, cross flow ultrafiltration was conducted under turbulent flow conditions in a rectangular channel (100 mm× 60 mm × 5.9 mm) using dextran as the solute. The cross flow velocities were 1.06, 1.84 and 2.75 m/s (Re = 11395, 19780 and 29,563). The transmembrane pressure drops were 200, 400 and 600 kPa. The feed concentrations were 0.43, 0.935 and 1.42 kg/m3 . The membrane permeability was 1.44 × 10−10 m3 /(N s). The flow rates and channel geometry

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of the experimental average permeate flux to that using various methods is presented in Fig. 12. It may be observed from the figure that the prediction of the present study with suction lies well within ±20% of the experimental values. Use of Dittus Boelter correlation grossly overpredicts the experimental results and use of film theory results into underprediction in most of the operating conditions, as expected. 5. Conclusions

Fig. 12. Variation of calculated data with experimental data [26].

used in these experiments ensure that the experiments were carried out in turbulent range. The proposed theory and simulation procedure presented in this paper are preformed with the same operating parameters and geometric configurations of their systems to make the theoretical procedure relevant to the experimental conditions reported in Ref. [26]. The comparison

Quantification of the effects of inclusion of suction in the mass transfer coefficient has been attempted in this work for the design of membrane modules under turbulent flow conditions and osmotic pressure controlled filtration. Simulation results are also compared with the conventional approach using film theory that assumes the constant thickness of mass transfer boundary layer at every differential element of the module. The calculations show that significant underprediction (about 70–75%) of the length averaged permeate flux is obtained without including suction in the mass transfer coefficient for various operating conditions. Using film theory, the underpredicion of the average

Fig. A1. Schematic diagram of numerical solution technique.

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permeate flux is obtained in the range of 30–35%. Using Dittus Boelter correlation for estimating mass transfer coefficient, the average permeate flux is predicted about 2.5 times that of the present study including suction in the mass transfer coefficient for a 10 m long membrane module. The calculated results are compared to the experimental data, available in the literature [26]. It is observed that the present study with suction matches the experimental data within ±20%, whereas, the film theory results into underprediction and use of DittusBoelter relationship results in substantial overprediction of the permeate flux. Appendix A A.1. Equations for film theory In this case, Eq. (18) will be of the following form,   cm − c p vw = k ln c − cp

(A1)

The algebraic equation for the estimation of membrane surface concentration (corresponding to Eq. (19)) becomes,   cm Rr 2 3 k ln = [P − (A1 cm + A2 cm + A3 cm )] c − cm (1 − Rr ) (A2) A.2. Dittus-Boelter correlation The Dittus-Boelter correlation is given as, Sh = 0.023(Re)0.8 (Sc)0.33

(A3)

A.3. Schematic of numerical scheme The schematic of numerical solution technique is presented in Fig. A1. It may be noted here that for calculation when suction effects are not incorporated in mass transfer coefficient, λ is set equal to zero and the rest of the calculations follow as shown in Fig. A1. References [1] W.F. Blatt, A. Dravid, A.S. Michaels, L. Nelson, Solute polarization and cake formation in membrane ultrafiltration: causes, consequences and control techniques, in: J.E. Flinn (Ed.), Membrane Science and Technology, Plenum Press, New York, 1970, p. 47. [2] C.W. van Oers, M.A.G. Vorstman, W.G.H.M. Muijselaar, P.J.A.M. Kerkhof, Unsteady state flux behaviour in relation to presence of a gel layer, J. Membr. Sci. 73 (1992) 231. [3] A.A. Kozinsky, E.N. Lightfoot, Protein ultrafiltration: a general example of boundary layer filtration, AIChE J. 34 (9) (1988) 1563. [4] H.M. Yeh, H.P. Wu, J.F. Dong, Effects of design and operating parameters on the declination of permeate flux for membrane ultrafiltration along hollow-fiber modules, J. Membr. Sci. 213 (2003) 33.

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