第4章 生物膜动力学

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利用以下2方程，通过计算传质通量，了解生物膜 General solution when S is w 中基质梯度的影响作用： 固定参数：D =1cm /d， X =40mg/cm ， K=0.001mg/cm ， known q=10mg/mg-d
f 2 f 3 3

ˆX 如果生物膜是厚的（ ：由第 d S Sw=0) q f f S f 2个方程： 0 D f 2-d for Jdeep=2.5mg/cm Ss=0.01mg/cm3; 2 dz Sw=0.005mg/cm K Sf 3,由第1个方程： 如果生物膜是薄的，
zLf
0
Df
ds f dz
z 0
Pseudoanalytical solution
dS D dz
z 0
see 10 references
How steady-state biofilm respond to changes in substrate concentration S:
The vertical axis also is proportional to the biomass accumulation per unit surface area, as XfLf=YJ/b’
Or:


L=D/km
D(Rem )0.75 Sc0.67 L 5.7u
km质量传递系数

The overall biofilm-loss coefficient
(It is related to the shear stress .)
b’=b+bdet
bdet=specific biofilm-来自百度文库etachment loss coefficient(T-1)
The equation requires two boundary conditions:
dS f dz
zLf
0
no flux into the attachment surface
At the biofilm/water interface, the external mass transport is described according to Fick’s first law:
2
J=1.9mg/cm2-d， 如果Sw=0.009mg/cm3， J=0.85mg/cm2-d 当生物膜边界两边SS和SW已知时， 1/ 2 方程解析解： K Sw ˆ J 2qX f D f ( S s S w K ln K S ) s
Integration of the equation yield closed-form •厚生物膜 Lf/ 1>1 analytical solutions for flux and Sf: Lf/1 <<1 。 •完全穿透生物膜
J1 D f S s tanh( L f / 1 )
1
f
Xf=50mg/cm3， K1=1.0cm3/mg-d （ 1=0.045 cm） Ss=0.1mg/cm3， J1=0.048mg/cm2-d； Case a： Df=6*10-5 cm2/d （ 1=0.0011 cm） J1=5.5*10-3 mg/cm2-d； or： K1Xf=8.3*105， J1=91 mg/cm2-d Therefore， a deep biofilm does not imply a large or a small flux， because a small 1 can be obtained from fast kinetics or from a low diffusion

K Sf The substrate is transported into the biofilm by molecular diffusion, give by Fick’s second law:
rut
ˆX f S f q
rdiff D f
d 2S f dz2
rdiff =rate of substrate accumulation due to diffusion(MsL-3T-1)
d ( X f dz) dt
Lf
Y
ˆS f q K Sf
Lf
( X f dz) b' X f dz
ˆS f q
Lf
0

0
d ( X f dz) dt


0
Y
K Sf
( X f dz)

0
b' X f dz
0 YJ b' X f Lf
稳态生物膜的基本条件： 单位面积上生物膜的生长 量与损失量平衡
For any position in the biofilm, the biomass is not at steady state, since Sf changes with position in the film.
生物膜表层和底部的净增长速率不同：
表层：d(Xfdz)/dt>0
底层： d(Xfdz)/dt<0
需要估计的参数：

Biomass density Xf: 5-200 mgVS/cm3, typical=40 mgVS/cm3 Diffusion coefficient in water, D (handbooks); Diffusion coefficient in the biofilm, Df=0.8D; Thickness of the effective diffusion layer:
For deep biofilm,Sw approaches zero, K ˆX f D f ( S s K ln Jdeep 2q leading to a very K S s useful result:
)
1/ 2
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The biofilm itself
S=0.11 S=0.07mg/cm
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1.for S<Smin, J=0, XfLf=0
2. S>Smin slightly, J , XfLf
3. J 的斜率从无穷变化到1，s＝0.07
4.对deep biofilm， J与XfLf无关， 额外的膜厚Sf=0。 5. 对于非常大的S，J=k1/2S1/211
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Average biofilm SRT

c
active
biom ass in of
the active
biofilm biom ass
production rate
1 c bdet X f L f Ab bdet
Ab=biofilm surface area
The key conclusion is that the average SRT for a steady-state biofilm is equal to the reciprocal of the specific detachment rate.
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rut
ˆX f S f q K Sf
The steady-state biofilm

虽然生物膜内部生物增长随所处位置不同有净增长和减少，但对生物膜整体来 讲，仍然可以应用稳态的概念，即单位面积上的生物膜量是常数（XfLf不随时 间变化)。即表示生物量的方程延生物膜总厚度积分等于零：
3
f For a steady-state concentration profile in theq biofilm. The substrate mass balance is:
Parameters: K,D ,D,L,
ˆ
0 Df
d 2S f dz 2

ˆX f S f q K Sf
Df:molecular diffusion coefficient of the substrate in the biofilm
第4章 生物膜动力学
Chapter 4 Biofilm Kinetics
Bruce E.Rittmann and Perry L. McCarty, Environmental Biotechnology:Principles and Applications. pp207-230
The Idealize biofilm 理想生物膜
D J (S S s ) L
求出：
同 时 求 解
J, XfLf（生物 膜积累量）， Lf q,K,Df,D,L,Y, b’,Xf 和 S 已
知时。
在参数
Active biomass in the biofilm:
0 YJ b' X f Lf
boundary conditions
dS f dz
Bulk layer liquid L
Diffusion Biofilm
•The biofolm has a uniform biomass density Xf(Mx/L) •It has a locally uniform thickness of Lf •Mass-transport resistance can be important inside the biofilm and to the biofilm
cosh((L f z ) / 1 ) cosh(L f / 1 )
J1=substrate flux into a first-order biofilm (MxL-2T-1)
1=Df/k1Xf standard biofilm depth dimension
S f Ss
tanh(x)=(ex-e-x)/(ex+e-x) cosh(x)=(0.5(ex+e-x))
ds f D J (S Ss ) D f L dz

z 0
dS D dz
J=substrate flux, M/J.T-1
z 0
D:molecular diffusion coefficient in water
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Illustration for First-order Kinetics
Full penetrated
Xf, density
（ Mx L-3）
shallow
s
s
s
ss
Deep
Lf,thickness
sf
Sw
Physical propertes
Characreristic concentration profiles
2
z
Substrate phenomena

At any position inside the biofilm,Substrate utilization:
0 Df d 2S f dz
2

ˆX f S f q K Sf
SfK
0 Df
d 2S f dz
2
k 1 XfSf
•标准深度是扩散速率于生物降解速率之比。 的深度。
•Lf/ 1 是无量纲生物膜厚度，它表示生物膜 k1=rate coefficient (L3Mx-1 T-1 )=q/K
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X f Lf
JY b' JY Lf X f b'
单位面积上生物膜量
生物膜厚度
Rittmann and McCarty(1980)
The steady-state –biofilm solution

Substrate mass balance equation: ˆX f S f d 2S f q 0 Df 2 dz K Sf Substrate tranport to the biofilm:

At a position inside the biofilm, a mass balance on active biomass is: ˆS f d ( X f dz) q '
生物膜单位面积上的变化量 （MxL-2T-1)
dt
Y
K Sf
( X f dz) b X f dz
dz=the thickness of a differential section of biofilm b’=an overall biofilm specific loss rate(T-1)
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The fellowing Fig. Shows concentration profiles for two biofilm having the same physical thickness(Lf=100m), K1,Xf,Df(1)对传质通量J1的影响： but distinctly different 1 values. Case b： D =0.1cm2/d，