Abstract
Ultrafiltration of Kraft black liquor was carried out by using an asymmetric membrane in a stirred batch cell, modified to work on a continuous mode. Spiegler-Kedem (SK) model from irreversible thermodynamics was used for the estimation of different membrane-solute parameters, like solute permeability (Pm) and reflection coefficient (Σ). The Pm andΣ so calculated from the above model were used to study the variation of these parameters with other process variables, like bulk concentration, pressure difference and stirrer speed. Finally, a simulation model was developed with the objective to predict permeate flux and rejection, which coupled the film theory, osmotic pressure model and SK model. The simulation results obtained from this study were validated with the experimental data using cellulose acetate membrane of 5,000 Da MWCO. Reasonably good agreements between the predicted and experimental values were observed and the average absolute deviation (AAD) for the prediction of flux and rejection using SK model was found to be 6.3%.
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Abbreviations
- AAD:
-
average absolute deviation [-]
- Aij :
-
element in the i-th row & j-th column position of the collocation matrix A [-]
- a1, a2, a3 :
-
model parameters of the SK model, defined by Eq. (8) [-]
- Bij :
-
element in the i-th row & j-th column position of the collocation matrix B [-]
- BL:
-
black liquor [-]
- C:
-
local solute concentration [kg·m-3]
- CA :
-
solute concentration [kg·m-3]
- Cb :
-
feed concentration [kg·-3]
- Cm :
-
membrane surface concentration [kg ·-3]
- Cp :
-
permeate concentration [kg·m-3]
- D:
-
diffusivity [m2 · s-1]
- JA :
-
solute flux through membrane [kg·m-2·s-1]
- Jv :
-
volumetric flux [m3·m-2·s-1]
- k:
-
mass transfer coefficient [m·s-1]
- MWCO:
-
molecular weight cut off [-]
- n:
-
speed of the stirrer [rps]
- N:
-
number of internal grid points in orthogonal collocation method[-]
- Pm :
-
local solute permeability per unit membrane thickness [m·s-1] (permeability coefficient)
- Ps :
-
local solute permeability in the membrane [m2·s-1]
- δP:
-
transmembrane pressure difference [Pa]
- Rm :
-
membrane hydraulic resistance [m-1]
- Ro :
-
observed rejection, (1-Cp/Cb), (dimensionless)
- Rr :
-
real rejection, (1-Cm/Cb), (dimensionless)
- r:
-
radius of membrane [m]
- S:
-
cross-sectional area of membrane [m2]
- SK:
-
Spieglar-Kadem [-]
- TMP:
-
transmembrane pressure [-]
- t:
-
time [s]
- UF:
-
ultrafiltration [-]
- V:
-
volume of filtrate collected [m3]
- x:
-
distance coordinate within the membrane with origin lying on the permeate side membrane surface [m]
- δx:
-
membrane thickness [m]
- z:
-
distance away from the membrane surface on feed side [m]
- δ:
-
film thickness [m]
- Μs :
-
solution viscosity [kg·m-1 ·s-1]
- Σ:
-
reflection coefficient, 0 for no rejection, 1 for total rejection
- Ω:
-
stirrer speed [s-1]
- π:
-
osmotic pressure [Pa
- δ:
-
difference [-]
- Μ:
-
viscosity [kg·m-1 · s-1]
- Ν:
-
kinematic viscosity, (Μ/ρ) [m2 · s-1]
- ρ:
-
density [kg·m-3]
- {ie623-01}:
-
function, defined by Eq. (11)
- b:
-
bulk
- i:
-
component
- m:
-
membrane surface
- p:
-
permeate phase
- j:
-
component
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Sarkar, P., Datta, S., Bhattacharjee, C. et al. Performance study on ultrafiltration of Kraft black liquor and membrane characterization using Spiegler-Kedem model. Korean J. Chem. Eng. 23, 617–624 (2006). https://doi.org/10.1007/BF02706804
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DOI: https://doi.org/10.1007/BF02706804