A biofilm extension of Freters model of a bioreactor with wall - - PowerPoint PPT Presentation

A biofilm extension of Freter’s model of a bioreactor with wall attachment and a failed attempt to optimize it

Hermann J. Eberl1 and Alma Maˇ si´ c2

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 = D

• S0 − S
• − γ−1

uµu(S) + δwµw(S)

• ˙

u = u

• µu(S) − D − ku
• + βδw + δwµw(S)
• 1 − G(W)
• − αu(1 − W)

˙ w = w

• µw(S)G(W) − β − kw
• + αu (1 − W) δ−1

with µu(S) = muS au + S , µw(S) = mwS aw + S , W = w wmax , G(W) = 1 − W 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(S0 − S) − uµu(S) γV − J(S, λ) V ˙ u = u (µu(S) − D − ku) + AρEλ2 − αu ˙ λ = v(λ, t) + αu Aρ − Eλ2 where λ: biofilm thickness: biofilm expansion due to microbial growth J(S, λ): substrate flux into biofilm (substrate consumption by biofilm) J(S, λ) = AdcC′(λ) v(λ, t): ”expansion velocity” of biofilm (biofilm growth) v(z, t) = z mλC Kλ + C − kλ

• dζ

(∗) C(z): substrate concentration in biofilm C′′ = ρmλ dCγ C Kλ + C , C′(0) = 0, C(λ) = S – 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(S0 − S) − 1 V uµu(S) γ + ADCj(S, λ)

• ˙

u = u (µu(S) − D − ku) + AρEλ2 − αu ˙ λ = γdc ρ j(λ, S) − kλλ + αu Aρ − Eλ2 where after integrating substrate BVP once j(λ, S) := ρ γdC λ µλ(C(z))dz – 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.

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• 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

(c)

• θ

Kλ tanh

• λ2θ

Kλ ≤ j(λ, S) ≤

• θ

Kλ+S tanh

• λ2θ

Kλ+S

(d) with θ := ρmλ/γdc we have Sθ Kλ + S ≤ ∂j ∂λ(0, S) ≤ Sθ Kλ

0.2 0.4 0.6 0.8 1 x 10

−3

5 10 15 x 10

4

λ (m) j1(λ,10) j(λ,10) j2(λ,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 (S0, 0, 0) exists for all parameters. It is asymptotically stable µu(S0) < D + ku + α and ∂j ∂λ(0, S0) < kλρ γdC and unstable if either µu(S0) > D + ku + α

• r

∂j ∂λ(0, S0) > kλρ γdC .

• Corollary. A sufficient condition for asymptotic stability of the trivial

equilibrium is µu(S0) < D + ku + α and S0 Kλ < kλ mλ . On the other hand, µu(S0) > D + ku + α

• r

S0 Kλ + S0 > kλ mλ is sufficient for instability.

H.J.Eberl - CSTR with Wall Attachment – 6

• Extension of Freter’s model for a biofilm reactor: analysis

1 2 2 4 x 10

8

Sin (g/m3) ∂ j/∂λ (g/m5) upper limit Sinθ/Kλ lower limit Sinθ/(Kλ+Sin) ∂ j/∂λ(0,Sin) kλρ/γDc STABLE UNSTABLE 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

20 40 60 80 100 0.5 1 1.5 2 2.5x 10

−3

D (1/day) suspended biomass (g) 20 40 60 80 100 0.05 0.1 0.15 0.2 0.25 D (1/day) biofilm biomass (g) Sin=10 Sin=7 Sin=4 Sin=10 Sin=7 Sin=4 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

0.5 1 1.5 2 4 6 8 10 12 14 16 area (m2) portion of substrate removal performed by suspended biomass (%) 0.5 1 1.5 1 2 3 4 5 6 area (m2) suspended biomass relative to total biomass (%) D=1 D=4 D=8 D=17 D=25 D=42 D=68 D=85 D=93 D=1 D=4 D=8 D=17 D=25 D=42 D=68 D=85 D=93

Summary: for small colonization area and flow rate, suspendeds can contribute substantially to substrate removal

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• 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 min

Q∈Ω

T

0 QSdt

T

• where Q : [0, Tmax] → I

R+

0 reactor flow rate, Ω specified later

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• 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

• ther 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: ⋄ scalarization by monotonic (linear) functionals F : I R2 → I R min

Q∈Ω F(Z(Q)) = min Q∈Ω ωβ

T QSdt + (1 − ω)T, 0 < ω < 1 ⋄ modified Pollack algorithm: For every T ∈ (Tmin, Tmax) solve min

Q∈Ω

T QSdt

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• Optimization: Optimal control problem in Bolza form

min

Q∈Ω wβ

T QSdt + (1 − w)T with Ω = {Q measureable, 0 ≤ Q ≤ Qmax} subject to ˙ S = Q V (S0 − S) − 1 V uµu(S) γ + ADCj(S, λ)

• ˙

u = u

• µu(S) − Q

V − ku

• + AρEλ2 − αu

˙ λ = γdc ρ j(λ, S) − kλλ + αu Aρ − Eλ2 ˙ Vb = −Q S(0) = 0, u(0) ≥ u0, λ(0) ≥ 0, Vb(0) = Vb,max

• - linear in control variable Q =

⇒ optimal control chatters

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• Optimization: Off-on functions

– look for optimal flow rate Q in the class of functions Q(t) = 0, for t < Tswitch

Vb,max T −Tswitch ,

for Tswitch ≤ t ≤ T and solve (using Pollack’s method) min

Tswitch,T

T

0 QSdt

T

• ,

s.t. 0 < Tmin ≤ Tswitch ≤ T ≤ Tmax

10 20 30 40 50 0.5 1 1.5

• bjective z2
• bjective z1

10 20 30 40 50 20 40 60 80 100 treatment time T (days) treated wastewater (%) 10 20 30 40 50 1 2 3 treatment time T (days) relative improvement in z1 with optimal

• ff−on fcn. vs. constant Q (%)
• const. Q
• ff−on Q

a) b) c)

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• Optimization: Off-on functions continued

– strong dependence on initial data:

5 10 15 20 1 2 3 4 5 6 7 8 treatment time T (days) relative improvement in z1 with off−on Q over constant Q (%) 5 10 15 20 2 4 6 8 10 12 treatment time T (days) relative improvement in z1 with off−on Q over constant Q (%) λ0=10 λ0=50 λ0=100 λ0=200 λ0=500 u0=0.005 u0=0.02 u0=0.05 u0=0.1 u0=0.5 a) b)

– 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 = Vb,max/T = const = ⇒ for all practical purposes, no control benefits

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• Optimization: Other approaches that we tried

0.5 1 0.05 0.1 time (days)

• ptimal Q (m3/day)

T=1 0.5 1 1.5 2 0.01 0.02 0.03 0.04 0.05 time (days)

• ptimal Q (m3/day)

T=2 1 2 3 0.01 0.02 0.03 0.04 time (days)

• ptimal Q (m3/day)

T=3 1 2 3 4 0.01 0.02 0.03 time (days)

• ptimal Q (m3/day)

T=4

2 4 6 8 0.004 0.008 0.012 0.016 time (days) Q(t) (m3/day) 100 nodes 50 nodes

– zero-max functions: divide [0, Tmax] into n subintervals of length ∆t = T/n and search for optimal Q : t → {0, Qmax} – an industry standard software package – a free academic software package that did not converge – all these approaches are computationally much more expensive than simple off-on functions – none performs better than simple off-on functions = ⇒ increased complexity does not give better solutions

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• Take home

– extended the Freter model for a bioreactor with wall attachment by combining it with a Wanner-Gujer style biofilm model (single species, single substrate) to assess contribution of suspended bacteria to sub- strate degradation in a biofilm reactor – model can formally be written as ODE, and qualitatively studied with elementary techniques – in biofilm reactors, at lower flow rates suspended bacteria can make a major contribution to substrate removal – at higher flow rates suspended are washed out – qualitative behaviour of model similar than simple Freter model, quan- titative big differences (did not have time to emphasize this) – multi-species setup will be essentially more complex: free boundary value problem for a coupled nonlocal parabolic-hyperbolic system (did not have time to cover this) – finite time treatment: optimization not worth the effort

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