A biofilm extension of Freter’s model of a bioreactor with wall attachment and a failed attempt to optimize it Hermann J. Eberl 1 and Alma Maˇ c 2 si´ 1 Dept. Mathematics and Statistics, University of Guelph 2 Center for Mathematics, Lund University supported by H.J.Eberl - CSTR with Wall Attachment – 0
• Freter’s model of a CSTR with wall attachment (since 1983) S 0 − S ˙ − γ − 1 � � � � S = D uµ u ( S ) + δwµ w ( S ) � � � � u = u ˙ µ u ( S ) − D − k u + βδw + δwµ w ( S ) 1 − G ( W ) − αu (1 − W ) + αu (1 − W ) δ − 1 � � w = w ˙ µ w ( S ) G ( W ) − β − k w with m u S m w S w G ( W ) = 1 − W µ u ( S ) = a u + S , µ w ( S ) = a w + S , W = , w max 1 . 1 − W S : substrate concentration u : unattached bacteria w : wall attached bacteria – major assumptions: ⋄ growth, lysis, attachment, detachment, washout of unattached cells ⋄ available wall space for attachment is limited ⋄ same substrate conditions for attached and unattached bacteria – studied in 1990s and 2000s by Smith, Ballyk, Jones, Kojouharov,... in this and extended versions (plug flow, etc): principle of competitive exclusion does not hold H.J.Eberl - CSTR with Wall Attachment – 1
• Extension of Freter’s model for a biofilm reactor: setup – wastewater treatment processes: activated sludge vs. biofilm processes – biofilm reactors are designed to provide ample surface for colonization (retention of biomass): Trickling Filters, Membrane Aerated Biofilm Reactors, Moving Bed Biofilm Reactors (MBBR) , etc – MBBR is an attempt to provide CSTR conditions for biofilms – due to biomass detachment suspended bacteria cannot be avoided; typ- ically not accounted for in design of biofilm processes – similar hybrids: IFAS (Integrated Fixed Film Activated Sludge) – limitation of the Freter model : in biofilm reactors wall attached bacteria develop in thick biofilms with substrate gradients = ⇒ het- erogeneous, spatially structured populations = ⇒ need to include a biofilm model for wall attached bacteria H.J.Eberl - CSTR with Wall Attachment – 2
• Extension of Freter’s model for a biofilm reactor: model S = D ( S 0 − S ) − uµ u ( S ) − J ( S, λ ) ˙ γV V u = u ( µ u ( S ) − D − k u ) + AρEλ 2 − αu ˙ λ = v ( λ, t ) + αu ˙ Aρ − Eλ 2 where λ : biofilm thickness: biofilm expansion due to microbial growth J ( S, λ ): substrate flux into biofilm (substrate consumption by biofilm) J ( S, λ ) = Ad c C ′ ( λ ) v ( λ, t ): ”expansion velocity” of biofilm (biofilm growth) � m λ C � z � v ( z, t ) = K λ + C − k λ dζ ( ∗ ) 0 C ( z ): substrate concentration in biofilm C ′′ = ρm λ C C ′ (0) = 0 , K λ + C , C ( λ ) = S d C γ – observe : v and J can be ”obtained” by integrating ( ∗ ) once H.J.Eberl - CSTR with Wall Attachment – 3
• Extension of Freter’s model for a biofilm reactor: analysis – formally re-write model as an ODE system S = D ( S 0 − S ) − 1 � uµ u ( S ) � ˙ + AD C j ( S, λ ) V γ u = u ( µ u ( S ) − D − k u ) + AρEλ 2 − αu ˙ λ = γd c ρ j ( λ, S ) − k λ λ + αu ˙ Aρ − Eλ 2 where after integrating substrate BVP once � λ ρ j ( λ, S ) := µ λ ( C ( z )) dz γd C 0 – ODE can be studied with elementary techniques – NOTE: evaluating R.H.S still requires to solve BVP!! Proposition. Initial value problem possess a unique, non-negative and bounded solution for all t > 0. We have either u ( t ) = λ ( t ) = 0 or u ( t ) > 0 , λ ( t ) > 0 for all t > 0. H.J.Eberl - CSTR with Wall Attachment – 4
• Extension of Freter’s model for a biofilm reactor: analysis Lemma (Properties of j ( λ, S ) ). For λ ≥ 0 , S ≥ 0 the function j ( λ, S ) is well-defined and differentiable. It has the following properties: (a) j ( · , 0) = j (0 , · ) = 0 (b) ∂j ∂S (0 , S ) = 0 � � � � λ 2 θ λ 2 θ θ θ (c) K λ tanh K λ ≤ j ( λ, S ) ≤ K λ + S tanh K λ + S (d) with θ := ρm λ /γd c we have K λ + S ≤ ∂j Sθ ∂λ (0 , S ) ≤ Sθ K λ 4 x 10 15 j 1 ( λ ,10) j( λ ,10) j 2 ( λ ,10) 10 5 0 0 0.2 0.4 0.6 0.8 1 λ (m) −3 x 10 H.J.Eberl - CSTR with Wall Attachment – 5
• Extension of Freter’s model for a biofilm reactor: analysis Proposition (stability of washout equilibrium). Washout equilib- rium ( S 0 , 0 , 0) exists for all parameters. It is asymptotically stable ∂λ (0 , S 0 ) < k λ ρ ∂j µ u ( S 0 ) < D + k u + α and γd C and unstable if either ∂λ (0 , S 0 ) > k λ ρ ∂j µ u ( S 0 ) > D + k u + α or . γd C Corollary. A sufficient condition for asymptotic stability of the trivial equilibrium is S 0 < k λ µ u ( S 0 ) < D + k u + α and . K λ m λ On the other hand, S 0 K λ + S 0 > k λ µ u ( S 0 ) > D + k u + α or m λ is sufficient for instability. H.J.Eberl - CSTR with Wall Attachment – 6
• Extension of Freter’s model for a biofilm reactor: analysis 8 x 10 upper limit S in θ /K λ lower limit S in θ /(K λ +S in ) ∂ j/ ∂λ (0,S in ) 4 k λ ρ / γ D c ∂ j/ ∂λ (g/m 5 ) STABLE UNSTABLE 2 0 0 1 2 S in (g/m 3 ) H.J.Eberl - CSTR with Wall Attachment – 7
• Extension of Freter’s model for a biofilm reactor: Simulations Steady state values of u , λ in dependence of dilution rate −3 2.5x 10 S in =10 S in =10 S in =7 S in =7 S in =4 S in =4 0.25 2 suspended biomass (g) 0.2 biofilm biomass (g) 1.5 0.15 1 0.1 0.5 0.05 0 0 0 20 40 60 80 100 0 20 40 60 80 100 D (1/day) D (1/day) H.J.Eberl - CSTR with Wall Attachment – 8
• Extension of Freter’s model for a biofilm reactor: Simulations Contribution of suspended biomass to substrate removal portion of substrate removal performed by suspended biomass (%) 16 6 D=1 D=1 D=4 D=4 suspended biomass relative to total biomass (%) 14 D=8 D=8 D=17 5 D=17 D=25 D=25 12 D=42 D=42 D=68 D=68 4 D=85 D=85 10 D=93 D=93 8 3 6 2 4 1 2 0 0 0 0.5 1 1.5 0 0.5 1 1.5 area (m 2 ) area (m 2 ) Summary: for small colonization area and flow rate, suspendeds can contribute substantially to substrate removal H.J.Eberl - CSTR with Wall Attachment – 9
• Optimization: setup – previous analysis is concerned with long term behaviour of the reactor in the case of continuous inflow of substrate – now: treat finite amount of substrate in finite time – can the process be optimized by controlling flow rate Q ? ⋄ treat as much substrate as possible ⋄ in as short a time as possible – vector optimization problem � � T � 0 QSdt min T Q ∈ Ω R + where Q : [0 , T max ] → I 0 reactor flow rate, Ω specified later H.J.Eberl - CSTR with Wall Attachment – 10
• Vector optimization – Edgeworth-Pareto optimality: a solution is optimal is further improve- ment of one objective is only possible at the expense of making the other one worse – enforces a trade-off between objectives – solution is not unique, typically infinitely many optima exist – solution can be represented graphically as Pareto front – convert vector optimization problem into a family of scalar problems: R 2 → I ⋄ scalarization by monotonic (linear) functionals F : I R � T Q ∈ Ω F ( Z ( Q )) = min min Q ∈ Ω ωβ QSdt + (1 − ω ) T, 0 < ω < 1 0 ⋄ modified Pollack algorithm : For every T ∈ ( T min , T max ) solve � T min QSdt Q ∈ Ω 0 H.J.Eberl - CSTR with Wall Attachment – 11
• Optimization: Optimal control problem in Bolza form � T Q ∈ Ω wβ min QSdt + (1 − w ) T 0 with Ω = { Q measureable , 0 ≤ Q ≤ Q max } subject to S = Q V ( S 0 − S ) − 1 � uµ u ( S ) � ˙ + AD C j ( S, λ ) V γ � µ u ( S ) − Q � + AρEλ 2 − αu u = u ˙ V − k u λ = γd c ρ j ( λ, S ) − k λ λ + αu ˙ Aρ − Eλ 2 ˙ V b = − Q S (0) = 0 , u (0) ≥ u 0 , λ (0) ≥ 0 , V b (0) = V b,max -- linear in control variable Q = ⇒ optimal control chatters H.J.Eberl - CSTR with Wall Attachment – 12
• Optimization: Off-on functions – look for optimal flow rate Q in the class of functions � 0 , for t < T switch Q ( t ) = V b,max T − T switch , for T switch ≤ t ≤ T and solve (using Pollack’s method) � � T � 0 QSdt min , s.t. 0 < T min ≤ T switch ≤ T ≤ T max T T switch ,T 1.5 100 a) b) c) relative improvement in z 1 with optimal 3 80 off−on fcn. vs. constant Q (%) treated wastewater (%) 1 objective z 1 60 2 40 0.5 1 20 const. Q off−on Q 0 0 0 0 10 20 30 40 50 0 10 20 30 40 50 0 10 20 30 40 50 treatment time T (days) objective z 2 treatment time T (days) H.J.Eberl - CSTR with Wall Attachment – 13
• Optimization: Off-on functions continued – strong dependence on initial data: 8 12 a) b) λ 0 =10 relative improvement in z 1 with off−on Q over constant Q (%) relative improvement in z 1 with off−on Q over constant Q (%) u 0 =0.005 λ 0 =50 u 0 =0.02 7 λ 0 =100 u 0 =0.05 10 λ 0 =200 u 0 =0.1 6 u 0 =0.5 λ 0 =500 8 5 4 6 3 4 2 2 1 0 0 0 5 10 15 20 0 5 10 15 20 treatment time T (days) treatment time T (days) – initial data typically not known = ⇒ optimum difficult to find – the less biomass initially in reactor the higher potential for control – overall very moderate compared to Q = V b,max /T = const = ⇒ for all practical purposes, no control benefits H.J.Eberl - CSTR with Wall Attachment – 14
Recommend
More recommend
