GLOBAL MODEL FOR OPTIMIZING CROSSFLOW MICROFILTRATION AND ULTRAFILTRATION PROCESSES

• US 20080017576A1
• Filed: 06/08/2007
• Published: 01/24/2008
• Est. Priority Date: 06/15/2006
• Status: Abandoned Application

• Current Assignee: Rensselaer Polytechnic Institute
• Original Assignee: Rensselaer Polytechnic Institute
• Inventors: Belfort, Georges, Baruah, Gautam, Venkiteshwaran, Adith
• Application Number:
  US11/760,407
• Publication Number:
  US 20080017576A1
• Time in Patent Office: Days
• US Class Current: 210/641
• CPC Class Codes: B01D 2315/12 Feed-and-bleed systems
• CPC Class Codes: B01D 2317/022 Reject series
• CPC Class Codes: B01D 2317/08 Use of membrane modules of different kinds
• CPC Class Codes: B01D 61/145 Ultrafiltration
• CPC Class Codes: B01D 61/147 Microfiltration
• CPC Class Codes: B01D 61/149 Multistep processes comprising different kinds of membrane processes selected from ultrafiltration or microfiltration
• CPC Class Codes: B01D 61/22 Controlling or regulating

Abstract

The present invention is a method for optimizing operating conditions for yield, purity, or selectivity of target species, and/or processing time for crossflow membrane filtration of target species in feed suspensions. This involves providing as input parameters: size distribution and concentration of particles and solutes in the suspension; suspension pH and temperature; physical and operating properties of membranes, and number and volume of reservoirs. The method also involves determining effective membrane pore size distribution; suspension viscosity, hydrodynamics, and electrostatics; pressure-independent permeation flux of the suspension and cake composition; pressure-independent permeation flux for each particle and overall observed sieving coefficient of each target species through cake deposit and pores; solving mass balance equations for all solutes; and iterating the mass balance equation for each solute at all possible permeation fluxes, thereby optimizing operating conditions. The invention also provides a computer readable medium for carrying out the method of the present invention.

Claims

1. A method for determining optimum operating conditions for yield of a target species, purity of a target species, selectivity of a target species and/or processing time for crossflow membrane filtration of a polydisperse feed suspension comprising one or more target solute or particle species, said method comprising:

providing as input parameters;

size distribution of the particles and solutes in the suspension, concentration of particles and solutes in the suspension, suspension pH and temperature, membrane thickness, membrane hydraulic permeability (Lp), membrane pore size or molecular weight cut off, membrane module internal diameter, membrane module length, membrane area, membrane porosity, filtration system configuration, and reservoir volume (V);

determining effective membrane pore size distribution (λ

′

), viscosity of the suspension, hydrodynamics of the suspension, electrostatics of the suspension, pressure-independent permeation flux (J_PD) of the suspension and cake composition, pressure-independent permeation flux [J_PI(i)] for each particle (i) in the suspension, and overall observed sieving coefficient of each target solute or particle species through cake deposit and pores of the membrane using said provided input parameters;

solving a solute mass balance equation for each target species in each reservoir of the feed suspension based on said provided size distribution of the particles and solutes in the suspension, concentration of particles and solutes in the suspension, suspension pH and temperature, membrane thickness, membrane hydraulic permeability, membrane pore size or molecular weight cut off, membrane module internal diameter, membrane module length, membrane area, membrane porosity, filtration system configuration, and reservoir volumes, and said determined effective membrane pore size distribution (λ

′

), viscosity of the suspension, hydrodynamics of the suspension, electrostatics of the suspension, pressure-independent permeation flux (JPD) of the suspension and cake composition, pressure-independent permeation flux [J_PI(i)] for each particle (i) in the suspension, and overall observed sieving coefficient of a particle through cake deposit and pores of the membrane; and

iterating the solute mass balance equation for each species at all possible permeation fluxes to determine purity, yield, selectivity, and/or processing time of crossflow filtration of the target species, thereby determining operating conditions that optimize for yield of a target species, selectivity of a target species, purity of a target species, and/or processing time for crossflow membrane filtration of a polydisperse feed suspension comprising one or more target solute or particle species.

2. The method according to claim 1, wherein said filtration system configuration comprises:

number of reservoirs in the filtration system;

number of membranes in the filtration system; and

connectivity of the filters and reservoirs.

3. The method according to claim 1, wherein said determining viscosity of the suspension is carried out using a modified Einstein-Smoluchowski equation:

η

/η

₀=1+2.5φ

_b+k₁φ

_b², wherein η

is bulk fluid viscosity (kg/m·

s) of the suspension, η

₀ is bulk fluid viscosity of the suspension without solute (kg/m·

s), k₁ is particle shape factor (−

), and φ

_b is particle volume fraction in the bulk suspension (−

).

4. The method according to claim 1, wherein said determining viscosity of the suspension is carried out by experimentation.

5. The method according to claim 1, wherein said determining effective membrane pore size distribution (λ

′

) is carried out using the equation;

λ

′

=1−

exp(−

a/2s), where s=(5η

δ

_mL_p/ε

₁)^1/2, a is solute particle size, η

is bulk fluid viscosity (kg/m·

s), δ

_m is membrane/cake thickness (m), L_p is hydraulic permeability of the membrane (m/s-Pa), and ε

₁ is cake/membrane porosity (−

)

6. The method according to claim 1, wherein said determining the hydrodynamics of the suspension comprises calculating wall shear rate as

$\gamma$

= 8 ⁢

V axial d ,

where V_axial is axial velocity in membrane bore (m/s) and d is internal diameter of membrane module bore (nm).

7. The method according to claim 1, wherein said determining the hydrodynamics of the suspension comprises:

calculating wall shear rate as obtained by

$\mathrm{\gamma <br><br>}=\frac{8<br><br>\text{\hspace{1em}<br><br>}<br><br>V_{\mathrm{axial}}}{d},$

where V_axial is obtained by back-calculation from a specified Reynold'"'"'s number (Re), where

$\mathrm{Re}=\frac{\mathrm{\rho <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>d<br><br>\text{\hspace{1em}<br><br>}<br><br>V_{\mathrm{axial}}}{{\mathrm{\eta <br><br>}}_0<br><br>\left(1+2.5<br><br>\text{\hspace{1em}<br><br>}<br><br>{\mathrm{\varphi <br><br>}}_b+k_1<br><br>{\mathrm{\varphi <br><br>}}_b^2\right)},$

where η

₀ is bulk fluid viscosity of the suspension without solute (kg/m·

s), k₁ is particle shape factor (−

), and φ

_b is particle volume fraction in the bulk suspension (−

).

8. The method according to claim 6, wherein the membrane is selected from the group consisting of a linear membrane and a shear-enhanced helical membrane.

9. The method according to claim 8, wherein the membrane is a shear-enhanced helical membrane.

10. The method according to claim 9, wherein said determining the hydrodynamics of the suspension further comprises multiplying γ

by 1.95 to obtain the wall shear rate.

11. The method according to claim 1, wherein said determining electrostatics of the suspension comprises:

determining pI and charge of each particle in the suspension;

selecting pH of the suspension;

selecting ionic strength of the suspension;

selecting the valency (Z) of ions in the suspension; and

obtaining the effective solute radius (a_effective) for each particle, using said determined pI and charge of each particle in the suspension, said selected pH and ionic strength of the suspension, and said valency (Z) of ions in the suspension, thereby determining the electrostatics of the suspension.

12. The method according to claim 11, wherein said obtaining the effective solute radius (a_effective) comprises calculating:

$a_{\mathrm{effective}}=a+\left(\frac{4<br><br>a^3<br><br>{\mathrm{\sigma <br><br>}}_s^2}{{\mathrm{\varepsilon <br><br>\varepsilon <br><br>}}_0<br><br>k^{\mathrm{\prime <br><br>}}<br><br>T}\right)<br><br>{\mathrm{\lambda <br><br>}}^{\mathrm{\prime <br><br>}}<br><br>\left(1-{\mathrm{\lambda <br><br>}}^{\mathrm{\prime <br><br>}}\right)<br><br>{\mathrm{\kappa <br><br>}}^{-1},$

where λ

′

is given as

${\mathrm{\lambda <br><br>}}^{\mathrm{\prime <br><br>}}=1-\exp <br><br>\left(\frac{-a}{2<br><br>s}\right);<br><br>$

κ

<sup>−

1</sup> is given as

${\mathrm{\kappa <br><br>}}^{-1}={\left(\frac{\mathrm{\varepsilon <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>\mathrm{RT}}{{\mathrm{Fa}}^2<br><br>\sum <br><br>Z_i^2<br><br>C_i}\right)}^{1/2};<br><br>$ ${\mathrm{\sigma <br><br>}}_s=\mathrm{no}.\text{\hspace{1em}<br><br>}<br><br>\mathrm{of}<br><br>\text{\hspace{1em}<br><br>}<br><br>\mathrm{charges}\times <br><br>\frac{e}{4<br><br>\mathrm{\pi <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>a^2},$

where colloids are assumed spherical, and wherein a is radius of species (m), k<sup>−

1</sup> is Boltzmann constant (J/mol K);

s is specific pore area (m);

ε

is permittivity of solvent (C²/J-m);

R is gas constant (J/mol-K);

T is temperature (K);

Fa is Faraday constant (C/mol);

Z_i is valency of ions;

C_i is concentration of ions (mol/m³);

σ

_s is surface charge (C/m²), and e is charge of one electron (C).

13. The method according to claim 11, wherein said determining pI and charge of each particle comprises using the Henderson-Hasselbach equation:

$p<br><br>\text{\hspace{1em}<br><br>}<br><br>H=p<br><br>\text{\hspace{1em}<br><br>}<br><br>K_a+\log <br><br>\left(\frac{\left[A\right]}{\left[\mathrm{HA}\right]}\right).$

14. The method according to claim 11, wherein said determining pI and charge of each particle is carried out using a computer readable program.

15. The method according to claim 11, wherein said selecting the pH of the suspension comprises:

choosing a pH that optimizes the yield, purity, selectivity, and/or diafiltration processing time of polydisperse suspensions and solutions or that is fixed by process requirements other than filtration.

16. The method according to claim 11, wherein said selecting ionic strength of the suspension comprises:

choosing an ionic strength that optimizes the yield, purity, selectivity, and/or diafiltration processing time of polydisperse suspensions and solutions or that is fixed by process requirements other than filtration.

17. The method according to claim 11, wherein said selecting the valency of ions (Z_i) in the suspension comprises choosing the (Z) value that optimizes the yield, purity, selectivity, and/or diafiltration processing time of polydisperse suspensions and solutions or that is fixed by process requirements other than filtration.

18. The method according to claim 1, wherein said determining the pressure-independent flux [J_PI(i)] for the polydisperse suspension and cake composition comprises:

1) determining the pressure-independent flux for a monodisperse suspension (J_mi) for a particle “

i”

using;

$J_{m<br><br>\text{\hspace{1em}<br><br>}<br><br>i}=\mathrm{Max}<br><br>\left[\mathrm{BD}<br><br>\text{\hspace{1em}<br><br>}<br><br>\ln <br><br>\left(\frac{{\mathrm{\varphi <br><br>}}_w}{{\mathrm{\varphi <br><br>}}_b}\right),\mathrm{SID}<br><br>\text{\hspace{1em}<br><br>}<br><br>\ln <br><br>\left(\frac{{\mathrm{\varphi <br><br>}}_w}{{\mathrm{\varphi <br><br>}}_b}\right)\right]$

where BD=0.114(γ

k′

²T²/n²a²L)^1/3, SID=0.078(a⁴/L)^1/3, and φ

_w=0.64 is set as maximum packing volume fraction for monodisperse spheres for each species for a first iteration;

2) determining maximum aggregate packing volume fraction for all particles (φ

_M) at the membrane wall using φ

_Mn=φ

_m+φ

_m(1−

φ

<sub>Mn−

1</sub>), where φ

_M=φ

_m is set to 0.64 when the size ratio of the particles is >

10, such that a_i+1>

10a_i for all a_i; and

φ

_M=φ

_m+φ

_m(1−

φ

_m)+0.74[1−

{φ

_m+φ

_m(1−

φ

_m)}]

3) iterating φ

_M for all particle sizes and selecting the particle that gives the minimum permeation flux at a given wall shear rate (J_PD), where (J_PD) is obtained by J_PD=Min[J_m1, J_m2, . . . , J_mn], where the selected particle has a radius α

_m;

4) determining packing density for other particle sizes (α

_i for i≠

m) at the minimum permeation flux by calculating φ

_wi from the equation;

${\mathrm{\varphi <br><br>}}_{\mathrm{wi}}=\mathrm{Min}<br><br>\left[{\mathrm{\varphi <br><br>}}_{\mathrm{bi}}<br><br>\exp <br><br>\left(\frac{J_{\mathrm{PD}}}{\mathrm{BD}}\right),{\mathrm{\varphi <br><br>}}_{\mathrm{bi}}<br><br>\exp <br><br>\left(\frac{J_{\mathrm{PD}}}{\mathrm{SID}}\right)\right]<br><br>\text{\hspace{1em}<br><br>}<br><br>\mathrm{for}<br><br>\text{\hspace{1em}<br><br>}<br><br>\mathrm{all}<br><br>\text{\hspace{1em}<br><br>}<br><br>i\ne <br><br>m;<br><br>$

5) checking Σ

φ

_wi≦

φ

_M and other packing constraints; and

6) determining a hypothetical pressure-independent flux [J_PI(i)] for each particle by;

$J_{\mathrm{PI}}<br><br>\left(i\right)=\mathrm{Max}<br><br>\left[\mathrm{BD}<br><br>\text{\hspace{1em}<br><br>}<br><br>\ln <br><br>\left(\frac{{\mathrm{\varphi <br><br>}}_{\mathrm{wi}}}{{\mathrm{\varphi <br><br>}}_{\mathrm{bi}}}\right),\mathrm{SID}<br><br>\text{\hspace{1em}<br><br>}<br><br>\ln <br><br>\left(\frac{{\mathrm{\varphi <br><br>}}_{\mathrm{wi}}}{{\mathrm{\varphi <br><br>}}_{\mathrm{bi}}}\right)\right],$

where φ

_wi=0.74(1−

Σ

φ

_wretained) using the results of steps

     1) to

     5), thereby determining pressure-independent permeation flux [J_PI(i)])] for the polydisperse suspension and cake composition of the suspension.

19. The method according to claim 18, wherein J_PI=J_PD for nominally retained particles.

20. The method according to claim 18, wherein J_PI≧

J_PD for transmitted particles.

21. The method according to claim 18, wherein said determining maximum aggregate packing volume fraction (φ

_M) at the membrane wall comprises;

calculating a maximum radius ratio of all particles;

determining if said maximum radius ratio is <

10; and

setting φ

_M as 0.68, where said maximum radius ratio is <

10.

22. The method according to claim 18 further comprising:

reevaluating the estimate of the pressure-independent polydisperse permeation flux of the suspension by correcting packing density using φ

_wicorrected=φ

_M[(φ

_wi)/Σ

φ

_wi] instead of 0.64; and

repeating steps

     1) and

     3).

23. The method according to claim 1 further comprising:

re-calculating packing density for all particle sizes if packing constraints are not satisfied based on initial determination of packing densities of the particles at the wall.

24. The method according to claim 1, wherein determining said overall observed sieving coefficient (S_o(i)) through the cake deposit and the membrane comprises:

using S_o(i)=S_odeposit(i)S_omem(i), where S_odeposit (sieving coefficient through the deposit) is

$S_{\mathrm{odeposit}}<br><br>\left(i\right)=1-\frac{J_{\mathrm{actual}}}{J_{\mathrm{PI}}<br><br>\left(i\right)}$

for the ith particle;

the sieving coefficient through the membrane S_omem(i) is obtained from

$S_{\mathrm{omem}}<br><br>\left(i\right)=\frac{S_a}{\left(1-S_a\right)<br><br>\exp <br><br>\left(\frac{-J_{\mathrm{actual}}}{k}\right)+S_a},\mathrm{where}$

mass transfer coefficient (k) is given by

$k=\frac{J_{\mathrm{PI}}<br><br>\left(i\right)}{\ln <br><br>\left(\frac{{\mathrm{\varphi <br><br>}}_{\mathrm{wi}}}{{\mathrm{\varphi <br><br>}}_{\mathrm{bi}}}\right)},$

where ø

_wi is particle volume fraction at the membrane wall (−

) for particle (i), ø

_bi is particle volume fraction in bulk solution (−

) for particle (i);

actual sieving coefficient (S_a) is obtained from

$S_a=\frac{S_{\mathrm{\infty <br><br>}}<br><br>\exp <br><br>\left({\mathrm{Pe}}_m\right)}{S_{\mathrm{\infty <br><br>}}+\exp <br><br>\left({\mathrm{Pe}}_m\right)-1},$

wall Peclet number (Pe_m) is obtained from

${\mathrm{Pe}}_m=\left(\frac{J_{\mathrm{actual}}<br><br>{\mathrm{\delta <br><br>}}_m}{D}\right)<br><br>\left(\frac{S_{\mathrm{\infty <br><br>}}}{\mathrm{\varepsilon <br><br>\varphi <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>K_d}\right),$

where φ

K_d=(1−

λ

′

)^9/2 and λ

′

is statistical equilibrium partition coefficient (−

); and

intrinsic sieving coefficient S_∞

is obtained by S_∞

=(1−

λ

′

)²[2−

(1−

λ

′

)²]exp(−

0.7146λ

′

²).

25. The method according to claim 1, wherein said solving a solute mass balance equation for each solute (i) comprises:

calculating the difference equation for each solute (i) using;

$\begin{array}{c}{\mathrm{\varphi <br><br>}}_{\mathrm{bi}<br><br>\text{\hspace{1em}<br><br>}<br><br>1}<br><br>\left(t+\mathrm{\Delta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>t\right)=<br><br>{\mathrm{\varphi <br><br>}}_{\mathrm{bi}<br><br>\text{\hspace{1em}<br><br>}<br><br>1}<br><br>\left(t\right)<br><br>\left[1-J<br><br>\left(1\right)<br><br>A<br><br>\left(1\right)<br><br>S_{o<br><br>\text{\hspace{1em}<br><br>}<br><br>1}<br><br>\left(i\right)<br><br>\frac{\mathrm{\Delta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>t}{V<br><br>\left(1\right)}\right]+\\ <br><br>{\mathrm{\varphi <br><br>}}_{\mathrm{bi}<br><br>\text{\hspace{1em}<br><br>}<br><br>2}<br><br>\left(t\right)<br><br>\left[J<br><br>\left(2\right)<br><br>A<br><br>\left(2\right)<br><br>S_{o<br><br>\text{\hspace{1em}<br><br>}<br><br>1}<br><br>\left(i\right)<br><br>\frac{\mathrm{\Delta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>t}{V<br><br>\left(1\right)}\right]\end{array}$

wherein A is membrane area (m²);

J is solvent permeation flux (m/s);

T is temperature (K);

V(1) is the volume of reservoir (1) (m³);

ø

_bi1 is the particle volume fraction in the bulk solution (−

) for solute particle i in a first reservoir;

S_o1(i) is overall observed sieving coefficient through the cake deposit and the membrane in a first reservoir.

26. The method according to claim 1, wherein said solving a solute mass balance equation for each solute (i) in each reservoir (j) comprises:

calculating the difference equation for each solute (i) for n reservoirs and n membranes using;

φ

_bij(t+Δ

t)=φ

_bij(t)+(1/V(j))[Σ

(k)φ

_bikS_ok(i)−

P_jφ

_bijS_oj(i)]Δ

t, wherein P(k) is permeation rate in m³/s through the kth membrane and wherein k=membrane numbers whose permeate is routed to reservoir (j) and k≠

j.

27. The method according to claim 1, wherein the operating conditions determined are optimum for yield of a target species from the crossflow filtration of particles in a polydisperse feed suspension.

28. The method according to claim 1, wherein the operating conditions determined are optimum for the purity of a target species from the crossflow filtration of particles in a polydisperse feed suspension.

29. The method according to claim 1, wherein the operating conditions determined are optimum for selectivity of a target species.

30. The method according to claim 1, wherein the operating conditions determined are optimum for processing time of the crossflow filtration of a target species in a polydisperse feed suspension.

31. The method according to claim 1, wherein crossflow filtration is carried out using ultrafiltration.

32. The method according to claim 1, wherein crossflow filtration is carried out using microfiltration.

33. The method according to claim 1, wherein crossflow filtration is carried out using ultrafiltration and microfiltration.

34. The method according to claim 1, wherein the feed suspension is selected from the group consisting of streams from biomedical and bio-processing industries, waste water, surface water, environmental pollutants, industrial waste streams, and industrial feed streams.

35. The method according to claim 34, wherein the feed suspension is a stream from biomedical and bio-processing industries selected from the group consisting of proteins, cells, nucleic acids, colloids, milk, and suspended particles.

36. The method according to claim 1, wherein optimum operating conditions for yield of a target species, purity of a solute, selectivity of a desired particle, or processing time crossflow membrane filtration of particles comprising one or more desired solutes in a polydisperse feed suspension are determined using a computer readable program.

37. The method according to claim 36, wherein time (t) is an arbitrarily small increment.

38. A computer readable medium having stored thereon programmed instructions for predicting and optimizing operating conditions for yield of a target species, purity of a target species, selectivity of a target species and/or processing time for crossflow membrane filtration of a polydisperse feed suspension comprising one or more target solute or particle species, said medium comprising:

a machine executable code which, when provided as input parameters;

size distribution of the particles and solutes in the suspension, concentration of particles and solutes in the suspension, suspension pH and temperature, membrane thickness, membrane hydraulic permeability (Lp), membrane pore size or molecular weight cut off, membrane module internal diameter, membrane module length, membrane area, membrane porosity, filtration system configuration, and reservoir volume (V); and

executed by at least one processor, causes the processor to calculate the effective membrane pore size distribution (λ

′

), viscosity of the suspension, hydrodynamics of the suspension, electrostatics of the suspension, pressure-independent permeation flux (J_PD) of the suspension and cake composition, pressure-independent permeation flux [J_PI(i)] for each particle (i) in the suspension, and overall observed sieving coefficient of each target solute or particle species through cake deposit and pores of the membrane using said provided input parameters; and

solve a solute mass balance equation for each target solute or particle species in each reservoir of the feed suspension based on said provided size distribution of the particles and solutes in the suspension, concentration of particles and solutes in the suspension, suspension pH and temperature, membrane thickness, membrane hydraulic permeability, membrane pore size or molecular weight cut off, membrane module internal diameter, membrane module length, membrane area, membrane porosity, filtration system configuration, and reservoir volumes, and said calculated effective membrane pore size distribution (λ

′

), viscosity of the suspension, hydrodynamics of the suspension, electrostatics of the suspension, pressure-independent permeation flux (JPD) of the suspension and cake composition, pressure-independent permeation flux [J_PI(i)] for each particle (i) in the suspension, and overall observed sieving coefficient of a particle through cake deposit and pores of the membrane;

iterate the solute mass balance equation for each species at all possible permeation fluxes to determine time, yield, selectivity, and processing time of crossflow filtration;

analyze the results of the mass balance equations and predict the operating conditions that optimize for yield of a target species, selectivity of a target species, purity of a target species, and/or processing time, thereby predicting and optimizing operating conditions for crossflow membrane filtration of a polydisperse feed suspension comprising one or more target solute or particle species.

39. A storage system containing the computer readable medium according to claim 38.