- Email: [email protected]
Diafiltration under condition of quasi-constant membrane surface concentration
Journal of Membrane Science 383 (2011) 301–302
Contents lists available at SciVerse ScienceDirect
Journal of Membrane Science journal homepage: www.elsevier.com/locate/memsci
Letter to the Editor Diafiltration under condition of quasi-constant membrane surface concentration Dear Sir, I have three comments to make concerning the recent diafiltration modelling paper by Paulen et al. [1] which introduced the useful parameter ˛ that was defined as the ratio of rate of water addition to rate of filtration. My first comment concerns the derivation of the macro-solute concentration (Cmax ) at which there is a maximum rate of removal of a low molecular weight component and the strategy for water addition. The second concerns assumptions, whose significance is such that they should have been highlighted. The third is a comment on definitions. Firstly the authors (with two major assumptions regarding the mass transfer coefficient k and the rejection coefficient R) confirmed the result obtained by Ng et al. [2] concerning the macro-solute concentration (Cmax ) which gives the maximum rate of removal of a low molecular weight component. This was found to be equal to Cg /e where e is the base of the natural logarithm and Cg is the quasi-constant membrane surface concentration, labelled in [1] as the gel concentration. The early paper [2] gives an approximate derivation of this result whilst the recent paper [1] obtains the result from application of control theory and by numerical analysis. It is now shown that this result can be obtained from mathematical analysis using a differential mass balance based upon the plant layout shown in Figure 1 of [1]. Consider a mass of micro-solute, ms , in a volume, V, that is being removed through diafiltration involving a membrane area, A. Let the rejection coefficient of the low molecular weight component be R and the flux J. The concentration of the micro-solute is signified by cs. It follows that dms AJ(1 − R)ms d(Vcs ) = = −AJ(1 − R)cs = − V dt dt
(1)
Hence 1 dms AJ(1 − R) d(ln(ms )) = = =− ms V dt dt
(2)
Now, as recognised by Paulen et al., the complete retention of the macro-solute means that V0 C0 = VC where 0 indicates initial conditions. Furthermore the flux is written as J = kln (Cg /C). Hence J and V can be eliminated from Eq. (2) to yield: A(1 − R)Ck ln(Cg /C) d(ln(ms )) =− V0 C0 dt
(3)
Given that A, V0 and C0 are constant, and with the assumption that k and R are also invariant with C, then differentiation with respect to C shows that the rate of removal of the micro-solute will be maximised at the value of C given by C = Cmax = Cg /e. This is true irrespective of the value of R provided it is constant. This transparent derivation confirms the result obtained in [1] and with some simple logic one can use this equation for the instantaneous rate of 0376-7388/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.memsci.2011.08.045
micro-solute removal to confirm the strategies they derived. There are four cases to consider depending upon (i) whether the initial concentration of macro-solute C0 is above or below Cg /e and (ii) whether the final target concentration of macro-solute Cf is above or below Cg /e. Subject to the two major assumptions on the mass transfer coefficient and on the rejection coefficient being reasonable, the four possible cases require the following strategies if time of operation is to be minimised.
Case 1 C0 > (Cg /e) ; Cf < Cg /e Step 1: dilute macro-solute concentration to Cg /e using ˛ = ∞ step; Step 2: remove micro-solute whilst maintaining C = Cg /e using UFCVD with ˛ = 1 to achieve desired removal of micro-solute; and Step 3: dilute macro-solute concentration to Cf using ˛ = ∞. Case 2 C0 > (Cg /e) ; Cf > Cg /e The closer the operation is to Cg /e the more rapid the removal rate for micro-solute and so as much of the operation as possible should occur at C = Cg /e. Step 1: dilute macrosolute concentration to Cg /e using ˛ = ∞ step; Step 2: remove micro-solute whilst maintaining C = Cg /e using UFCVD with ˛ = 1 until the point at which time for the final amount of micro-solute removal equals the time to achieve re-concentration of macrosolute from Cg /e to Cf ; so Step 3: pure UF with ˛ = 0. Case 3 C0 < (Cg /e) ; Cf > Cg /e The concentrations of macro-solute are either side of C = Cg /e and so the strategy is to reach this concentration as rapidly as possible and to remove micro-solute whilst maintaining C = Cg /e using UFCVD with ˛ = 1 until the point at which time for the final amount of micro-solute removal equals the time to achieve re-concentration of macro-solute from Cg /e to Cf . Hence Step 1: pure UF with ˛ = 0; Step 2: UFCVD with ˛ = 1 and Step 3: pure UF with ˛ = 0. Case 4 C0 < (Cg /e) ; Cf < Cg /e As dilution from Cg /e to Cf can be achieved theoretically in an instant using ˛ = ∞ the strategy is to reach the concentration Cg /e as rapidly as possible and to remove micro-solute using UFCVD with ˛ = 1 and to finish with pure dilution. Hence Step 1: pure UF with ˛ = 0; Step 2: UFCVD with ˛ = 1 and Step 3: pure dilution with ˛ = ∞.
Turning to my second observation, we start my noting that in [1] it is stated that the findings of the work have to be interpreted in the light of the six assumptions made at the end of Section 2. Unfortunately the assumption that the rejection coefficient of the micro-solute is assumed to be zero, which is mentioned previously in Section 2, is neither mentioned in this list nor in the conclusions. Mention is also omitted from the abstract. This is important as the routes shown in the state diagrams in Figures 4 and 5 of [1] are only true for R = 0 and this restriction might be missed by many readers. Also the assumption of constant mass transfer coefficient is crucial to the results and, in my opinion, should have been given prominence. Indeed this limitation makes the outcome C = Cmax = Cg /e of questionable significance.
302
Letter to the Editor / Journal of Membrane Science 383 (2011) 301–302
The final observation is that the membrane community has been correct to have reservations about the gel polarisation theory and this should not be passed over when choosing paper titles and symbols. As with others (e.g. [3]), it is recognised that the form of the “gel” polarization model provides a simple and convenient procedure for interpreting experimental UF data but it is important to make a distinction between film theory for mass transfer as applied to UF and gel polarisation theory. Paulen et al. [1] incorrectly state that Ng et al. [2] used the latter whereas they clearly used the former in combination with osmotic pressure considerations. The limiting flux generally occurs independently of any supposed gelation effects, and this is indeed recognised by Paulen et al., who correctly noted that fitting procedures often result in a physically unreasonable gelation value for the membrane surface concentration. Thus it is regrettable that the term “gel” concentration has not been replaced by a more suitable term and is indeed part of the title of the paper. Whilst the term “apparent gel” concentration would be more appropriate, it is further suggested, in common with many others (e.g. [2,3]), that Cw and not Cg should be adopted to signify the quasi-constant membrane surface concentration. Yours faithfully,
References [1] R. Paulen, et al., Minimizing the process time for ultrafiltration/diafiltration under gel polarization conditions, J. Membr. Sci. (2011), doi:10.1016/j.memsci.2011.06.044. [2] P. Ng, J. Lundblad, G. Mitra, Optimization of solute separation by diafiltration, Sep. Sci. 11 (5) (1976) 499–502. [3] P. Aimar, R. Field, Limiting flux in membrane separations: a model based on the viscosity dependency of the mass transfer coefficient, Chem. Eng. Sci. 47 (3) (1992) 579–586.
Robert Field ∗ Department of Engineering Science, University of Oxford, Parks Road, Oxford OX1 3PJ, UK ∗ Corresponding
author. Tel.: +44 1865 273814; fax: +44 1865 273905. E-mail address: robert.fi[email protected] 2 August 2011 Available online 26 August 2011
Contents lists available at SciVerse ScienceDirect
Journal of Membrane Science journal homepage: www.elsevier.com/locate/memsci
Letter to the Editor Diafiltration under condition of quasi-constant membrane surface concentration Dear Sir, I have three comments to make concerning the recent diafiltration modelling paper by Paulen et al. [1] which introduced the useful parameter ˛ that was defined as the ratio of rate of water addition to rate of filtration. My first comment concerns the derivation of the macro-solute concentration (Cmax ) at which there is a maximum rate of removal of a low molecular weight component and the strategy for water addition. The second concerns assumptions, whose significance is such that they should have been highlighted. The third is a comment on definitions. Firstly the authors (with two major assumptions regarding the mass transfer coefficient k and the rejection coefficient R) confirmed the result obtained by Ng et al. [2] concerning the macro-solute concentration (Cmax ) which gives the maximum rate of removal of a low molecular weight component. This was found to be equal to Cg /e where e is the base of the natural logarithm and Cg is the quasi-constant membrane surface concentration, labelled in [1] as the gel concentration. The early paper [2] gives an approximate derivation of this result whilst the recent paper [1] obtains the result from application of control theory and by numerical analysis. It is now shown that this result can be obtained from mathematical analysis using a differential mass balance based upon the plant layout shown in Figure 1 of [1]. Consider a mass of micro-solute, ms , in a volume, V, that is being removed through diafiltration involving a membrane area, A. Let the rejection coefficient of the low molecular weight component be R and the flux J. The concentration of the micro-solute is signified by cs. It follows that dms AJ(1 − R)ms d(Vcs ) = = −AJ(1 − R)cs = − V dt dt
(1)
Hence 1 dms AJ(1 − R) d(ln(ms )) = = =− ms V dt dt
(2)
Now, as recognised by Paulen et al., the complete retention of the macro-solute means that V0 C0 = VC where 0 indicates initial conditions. Furthermore the flux is written as J = kln (Cg /C). Hence J and V can be eliminated from Eq. (2) to yield: A(1 − R)Ck ln(Cg /C) d(ln(ms )) =− V0 C0 dt
(3)
Given that A, V0 and C0 are constant, and with the assumption that k and R are also invariant with C, then differentiation with respect to C shows that the rate of removal of the micro-solute will be maximised at the value of C given by C = Cmax = Cg /e. This is true irrespective of the value of R provided it is constant. This transparent derivation confirms the result obtained in [1] and with some simple logic one can use this equation for the instantaneous rate of 0376-7388/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.memsci.2011.08.045
micro-solute removal to confirm the strategies they derived. There are four cases to consider depending upon (i) whether the initial concentration of macro-solute C0 is above or below Cg /e and (ii) whether the final target concentration of macro-solute Cf is above or below Cg /e. Subject to the two major assumptions on the mass transfer coefficient and on the rejection coefficient being reasonable, the four possible cases require the following strategies if time of operation is to be minimised.
Case 1 C0 > (Cg /e) ; Cf < Cg /e Step 1: dilute macro-solute concentration to Cg /e using ˛ = ∞ step; Step 2: remove micro-solute whilst maintaining C = Cg /e using UFCVD with ˛ = 1 to achieve desired removal of micro-solute; and Step 3: dilute macro-solute concentration to Cf using ˛ = ∞. Case 2 C0 > (Cg /e) ; Cf > Cg /e The closer the operation is to Cg /e the more rapid the removal rate for micro-solute and so as much of the operation as possible should occur at C = Cg /e. Step 1: dilute macrosolute concentration to Cg /e using ˛ = ∞ step; Step 2: remove micro-solute whilst maintaining C = Cg /e using UFCVD with ˛ = 1 until the point at which time for the final amount of micro-solute removal equals the time to achieve re-concentration of macrosolute from Cg /e to Cf ; so Step 3: pure UF with ˛ = 0. Case 3 C0 < (Cg /e) ; Cf > Cg /e The concentrations of macro-solute are either side of C = Cg /e and so the strategy is to reach this concentration as rapidly as possible and to remove micro-solute whilst maintaining C = Cg /e using UFCVD with ˛ = 1 until the point at which time for the final amount of micro-solute removal equals the time to achieve re-concentration of macro-solute from Cg /e to Cf . Hence Step 1: pure UF with ˛ = 0; Step 2: UFCVD with ˛ = 1 and Step 3: pure UF with ˛ = 0. Case 4 C0 < (Cg /e) ; Cf < Cg /e As dilution from Cg /e to Cf can be achieved theoretically in an instant using ˛ = ∞ the strategy is to reach the concentration Cg /e as rapidly as possible and to remove micro-solute using UFCVD with ˛ = 1 and to finish with pure dilution. Hence Step 1: pure UF with ˛ = 0; Step 2: UFCVD with ˛ = 1 and Step 3: pure dilution with ˛ = ∞.
Turning to my second observation, we start my noting that in [1] it is stated that the findings of the work have to be interpreted in the light of the six assumptions made at the end of Section 2. Unfortunately the assumption that the rejection coefficient of the micro-solute is assumed to be zero, which is mentioned previously in Section 2, is neither mentioned in this list nor in the conclusions. Mention is also omitted from the abstract. This is important as the routes shown in the state diagrams in Figures 4 and 5 of [1] are only true for R = 0 and this restriction might be missed by many readers. Also the assumption of constant mass transfer coefficient is crucial to the results and, in my opinion, should have been given prominence. Indeed this limitation makes the outcome C = Cmax = Cg /e of questionable significance.
302
Letter to the Editor / Journal of Membrane Science 383 (2011) 301–302
The final observation is that the membrane community has been correct to have reservations about the gel polarisation theory and this should not be passed over when choosing paper titles and symbols. As with others (e.g. [3]), it is recognised that the form of the “gel” polarization model provides a simple and convenient procedure for interpreting experimental UF data but it is important to make a distinction between film theory for mass transfer as applied to UF and gel polarisation theory. Paulen et al. [1] incorrectly state that Ng et al. [2] used the latter whereas they clearly used the former in combination with osmotic pressure considerations. The limiting flux generally occurs independently of any supposed gelation effects, and this is indeed recognised by Paulen et al., who correctly noted that fitting procedures often result in a physically unreasonable gelation value for the membrane surface concentration. Thus it is regrettable that the term “gel” concentration has not been replaced by a more suitable term and is indeed part of the title of the paper. Whilst the term “apparent gel” concentration would be more appropriate, it is further suggested, in common with many others (e.g. [2,3]), that Cw and not Cg should be adopted to signify the quasi-constant membrane surface concentration. Yours faithfully,
References [1] R. Paulen, et al., Minimizing the process time for ultrafiltration/diafiltration under gel polarization conditions, J. Membr. Sci. (2011), doi:10.1016/j.memsci.2011.06.044. [2] P. Ng, J. Lundblad, G. Mitra, Optimization of solute separation by diafiltration, Sep. Sci. 11 (5) (1976) 499–502. [3] P. Aimar, R. Field, Limiting flux in membrane separations: a model based on the viscosity dependency of the mass transfer coefficient, Chem. Eng. Sci. 47 (3) (1992) 579–586.
Robert Field ∗ Department of Engineering Science, University of Oxford, Parks Road, Oxford OX1 3PJ, UK ∗ Corresponding
author. Tel.: +44 1865 273814; fax: +44 1865 273905. E-mail address: robert.fi[email protected] 2 August 2011 Available online 26 August 2011