Mathematical modelling of a waste water filtration process based on membrane filters Matt Hennessy, Bolor Jargalsaikhan, Diego Passarella, Juan Jos´ e Silva Torres, Clara Villar Marco with Iacopo Borsi IV UCM Modelling Week Madrid – June 14-22, 2010 Group 1 Modelling of Filtration Membranes
Outline Introduction to the problem 1 Averaging process 2 Scaling of the 1D system 3 Numerical Simulation 4 Conclusions 5 Group 1 Modelling of Filtration Membranes
As a reminder, three porosity approach Subscripts notation : ( · ) c is referred to the capillary region. ( · ) m is referred to the membrane region. ( · ) is referred to the shell region. Main tasks of our project: Modelling and simulation of filtration process Optimization of the parameters of the filters Group 1 Modelling of Filtration Membranes
The complete system ∇ · q c = − α c ( c m ) k m µl ( P c − P m ) ∇ · q m = α c ( c m ) k m µl ( P c − P m ) − αk m l ( P m − P ) ∇ · q = αk m µl ( P m − P ) � � ∂c α c ( c m ) k m ε c ∂t + ∇ · ( c q c ) = ε c ∇ · ( D ∇ c ) − γ µl ( P c − P m ) ( ε c c ) � � ∂c m α c ( c m ) k m = γ µl ( P c − P m ) ( ε c c ) ∂t 1 α c ( c m ) = A v . 1 + c m /c ref Group 1 Modelling of Filtration Membranes
Boundary conditions On the inlet boundary: q c · n = − J in . q m · n = 0 . q · n = 0 . c = c in . On the outlet boundary: q c · n = 0 . q m · n = 0 . q · n = J out . No flux condition for c . Elsewhere: no flux condition for both the hydrodynamic and the transport problem. Group 1 Modelling of Filtration Membranes
Averaging How could we reduce our 3D problem? We define the mean value: 2 π R 1 � � � F � ( z, t ) ≡ F ( x, y, z, t ) r d r d θ πR 2 0 0 We use, � � ∇ · F d V = F · n d S V s We suppose, � α c ( c m ) · ( P c − P m ) � ≈ � α c ( c m ) � · � ( P c − P m ) � Group 1 Modelling of Filtration Membranes
Getting the 1D problem ∂ 2 P c ∂z 2 = − α c ( c m ) k m − k c l ( P c − P m ) ∂ 2 P m = α c ( c m ) k m l ( P c − P m ) − αk m − k m l ( P m − P ) ∂z 2 ∂ 2 P ∂z 2 = αk m µ Q in − k z l ( P m − P ) − χ ( z )(2 R out ) πR 2 A out ∂z ( c q c ) = ε c D ∂c 2 ∂t + ∂ ∂c � α c ( c m ) k m � ε c ∂z 2 − γ µl ( P c − P m ) ( ε c c ) ∂c m � α c ( c m ) k m � = γ µl ( P c − P m ) ( ε s c ) ∂t 1 α c ( c m ) = A v , 1 + c m /c ref Group 1 Modelling of Filtration Membranes
Getting the 1D problem (cont.) + B.C. and I.C. On the inlet boundary ( z = 0) : q c = − J in q m = 0 = q c = c in . On the outlet boundary ( z = L ) : No flux condition for all the eq.s Group 1 Modelling of Filtration Membranes
The dimensionless form: � k c P ∗ � ∂ ( cq c ) 1 ∂c ∂t + = T c ε c µL 2 ∂z � D � ∂ 2 c � k m P ∗ � � 1 � = ∂z 2 − γA v ( P c − P m ) c L 2 µl 1 + c m /c ref Define: t adv = ε c µL 2 t diff = L 2 µl t attach = 1 k c P ∗ , D , t filt = A v k m P ∗ , γ t filt . Remark: T c T c = T filt ∼ O (10 3 ) s ; t diff ∼ O (10 5 ) s = ⇒ ≪ 1 t diff Therefore, the diffusion is negligible . Group 1 Modelling of Filtration Membranes
Our request: ⇒ ε c µL 2 µl t adv ≪ t filt = k c P ∗ ≪ A v k m P ∗ , Φ := ε c L 2 k m A v ≪ 1 . k c l Substituting the definition of k c , A v and ε c , we have the following condition: Φ = 16 k m L 2 ≪ 1 . lr 3 i Notice that: Φ depends only on the filter and the membrane parameters (no dependence upon the process). Φ depends on r i but not on r o ; that’s a good point, since the pollutant flows only through the capillary region. Group 1 Modelling of Filtration Membranes
Different Approaches: Simplified model using Matlab 1 Comsol Multiphysics software built-in models 2 Group 1 Modelling of Filtration Membranes
A simplified approach Additional scaling leads to the following non-dimensional equations for the pressures ∂ 2 p c ∂z 2 = θ c α c ( c m )( p c − p m ) , ∂ 2 p m = θ m [ β ( p m − p s ) − α c ( c m )( p c − p m )] , ∂z 2 ∂ 2 p s ∂z 2 = θ s ( p s − p m ) + ζχ ( z ) , and for the concentrations ∂ 2 c ∂t + τ filt ∂c ∂z ( cq c ) = τ filt ∂ ∂z 2 − γα c ( c m )( p c − p m ) c, τ adv τ diff ∂c m = γα c ( c m )( p c − p m ) c. ∂t Group 1 Modelling of Filtration Membranes
Leading order equations Pressure equations become much simpler ∂ 2 p c ∂z 2 = θ c α c ( c m )( p c − p m ) , 0 = β ( p m − p s ) − α c ( c m )( p c − p m ) , d 2 p s d z 2 = 0 . Concentration equation now hyperbolic ∂c ∂t + τ filt ∂ ∂z ( cq c ) = − γα c ( c m )( p c − p m ) c, τ adv ∂c m = γα c ( c m )( p c − p m ) c, ∂t where the capillary flux is given by q c = − ∂p c ∂z − p h Group 1 Modelling of Filtration Membranes
First implementation Even the simplified system cannot be solved completely by hand Implement a simple numerical scheme in Matlab Idea is to decompose the problem into two smaller problems; one for the pressure and one for the concentration Outline of algorithm: Assume concentration at time-step i is known (initial condition, for 1 example) Solve an elliptic equation for the capillary pressure at time-step i 2 Use new pressures to advance concentrations in time 3 Repeat 4 Group 1 Modelling of Filtration Membranes
Further numerical details Spatial discretizations using finite differencing First order upwinding used for concentration equation Elliptic equation for capillary pressure linear, solved using \ Time-stepping handled using ode15s Group 1 Modelling of Filtration Membranes
Results: attached matter 0.05 0.045 0.012 0.04 0.01 0.035 0.03 0.008 0.025 t 0.006 0.02 0.015 0.004 0.01 0.002 0.005 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 z Group 1 Modelling of Filtration Membranes
Results: trans-membrane pressure 40 Experimental Numerical 35 30 Trans-membrane pressure [kPa] 25 20 15 10 5 0 0 0.05 0.1 0.15 0.2 0.25 t [min] Group 1 Modelling of Filtration Membranes
Comsol Approach (I) Earth Science Module was used Application Modes: Darcy’s law for pressures (3 pde’s) Solute Transport for transport of pollutant (2 pde’s) Transient analysis for the whole system ( t ∈ [0 , 1] ) Process made of several cycles Each cycle has two stages, filtration (F) and backwash (BW) Matlab scripting to reproduce filtration-backwash cycles Physical parameters calibrated for one F-BW cycle Group 1 Modelling of Filtration Membranes
Comsol Approach (II) Darcy’s law application mode for p c , p m and p : � � δ S S ∂p κ ∂t + ∇ · − δ κ η ∇ ( p + ρ f g D ) = δ Q Q S S : Storage coefficient ( S = 10 − 4 ⇒ pseudo-stationary problem) p : Pressure in the porous media κ : Permeability η : Dynamic viscosity ρ f : Fluid density D : Vertical elevation Q S : Flow source/sink δ S,κ,Q : Scaling coefficients ( δ S = 1 /τ filt or 1 /τ back for filtration/backwash) Group 1 Modelling of Filtration Membranes
Comsol Approach (III) Solute Transport application mode for c and c m : ∂c δ ts 1 θ s ∂t + ∇ · ( − θ s D L ∇ c ) = − u · ∇ c + S c δ ts 1 : Time scaling coefficient ( = 1 /τ filt or 1 /τ back for filtration/backwash) θ s : Porosity c : Solute concentration D L : Diffusion coefficient ( 10 − 5 for numerical stability) u : Darcy’s velocity ( u = 0 for c m ) S c : Solute source/sink o.d.e. for c m is solved as a diffusion equation with very low diffusivity Group 1 Modelling of Filtration Membranes
Comsol Approach (IV) Matlab scripting for process simulation: One main file defines all parameters, cycles and function calling Two functions are called to solve F and BW stages Each function solves the 5 pde’s according to I.C. and B.C. of each stage of the cycles c f = i ( x,t = T bw ) = c f = i +1 ( x,t = T f ) = c bw = i ( x,t =0) , c bw = i ( x,t =0) , ... Concentrations are averaged along the filter after each stage (assumed constant, see Φ parameter) Group 1 Modelling of Filtration Membranes
Comsol Approach (V) Matlab scripting for process simulation (cont.): Filter parameters ( κ m , γ ) calibrated to fit experimental data ( TMP ≃ ( P in + P out ) / 2 ) Once parameters are calibrated, the process is optimized depending on τ filt , τ back and number of cycles Goal: maximize the ratio purified/used water of the process for given operation conditions Group 1 Modelling of Filtration Membranes
Comsol Approach (VI) Parameter fitting depending on κ m , γ and operation condition: 40 20 0 TMP (KPa) −20 −40 −60 Simulation Data −80 −100 0 0.05 0.1 0.15 0.2 0.25 Time (min) Group 1 Modelling of Filtration Membranes
Final Remarks Averaging leads to a set of simplified equations Not simple enough to solve by hand Two different numerical approaches were implemented Using MATLAB: Simplest case, but unable to match experimental 1 data Using COMSOL: Filtration data could be reproduced. Not enough 2 data to compare backwash stage Future work: Explore higher order behaviour in the simple model Model pore adsorption in the membrane, reduction in permeability Spatially dependent filter properties (permeability, etc.) Simulate filtration/backwash process over several cycles Group 1 Modelling of Filtration Membranes
Thank you for your attention Group 1 Modelling of Filtration Membranes
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