Development of a Dynamic Mathematical Model for Membrane Bioelectrochemical Reactors with Different Configurations

Jian Liand Zhen He *

Department of Civil and Environmental Engineering, Virginia Polytechnic Institute and State University, Blacksburg, VA 24060, USA

Supplementary data

* Corresponding author: phone (540) 231-1346; email:

Table S1(a). Parameters of the MBER-1 for electrical performance

Parameters / Description / Value / Units
F / Faraday constant / 96,485 / C mol-1
R / Ideal gas constant / 8.314 / J K-1 mol-1
T / Temperature / 298.15 / K
Y / Yield of anodophilic / 10.5 / mg-M mg-S-1
YCH4 / Methane Yield / 0.3 / mL CH4 mg-S-1
qmax,a / Maximum anodophilic reaction rate / 2.32 / mg-S mg-x-1 d-1
qmax,m / Maximum methanogenic reaction rate / 8.2 / mg-S mg-x-1 d-1
µmax, a / Maximum anodophilic growth rate / 0.797 / d-1
µmax, m / Maximum methanogenic growth rate / 0.1 / d-1
Ks,a / Half-rate constant of anodophilics / 20 / mg-S L-1
Ks,m / Half-rate constant of methanogens / 80 / mg-S L-1
m / Electrons transfer per mol of mediator / 2 / mol-1 mol mediator -1
γ / Mediator molar mass / 663,400 / mg-M mol mediator-1
Mtotal / Mediator fraction / 0.05 / mg-M mg-x-1
KM / Mediator half rate constant / 0.01 / mg-M L-1
Kd,a / Decay rate of anodophilc microorganism / 0.02*µmax,a / d-1
Kd,m / Decay rate of methanogenic microorganism / 0.02*µmax,m / d-1
Xmax,a / Anodophilc biofilm space limitation / 512.5 / mg-x L-1
Xmax,m / Methanogenic biofilm space limitation / 525 / mg-x L-1
Kx / Steepness Factor / 0.04 / L mg-x-1
Rmin / Minimum internal resistance / 31 / Ω
Rmax / Maximum internal resistance / 2000 / Ω
Emin / Minimum open circuit potential / 0.1 / V
Emax / Maximum open circuit potential / 0.7 / V
KR / steepness factor correction coefficient / 0.0818 / L mg-x-1

Table S1(b). Parameters of the MBER-1 for membrane performance

Parameters / Description / Value / Units
η / Water apparent viscosity / 0.00089 / Pa.S
A / Membrane surface area / 0.0051 / m-2
ρ / Specific cake resistance / 100000000 / m g-1
γ / Coefficient of β / 0.05
Y / Yield coefficient of the substrate consumption / 1.4 / mg-Mmg-S-1
µs,max / Maximum growth rate / 0.09 / d-1
Jair / Air crossflow / 80 / m3 m-2 d-1
Ks / Half saturation of substrate / 10 / g m-3
Kair / Half saturation of airflow / 4.6*10-5 / g m-3
β / Resistance of detachable cake by air crossflow / 0.01 / m-1
Q / Inflow / 0.000336 / m3 d-1
V / Volume of cathodic compartment / 0.00022 / m3

Table S2(a). Parameters of the MBER-2 for electrical performance

Parameters / Description / Value / Units
F / Faraday constant / 96,485 / C mol-1
R / Ideal gas constant / 8.314 / J K-1 mol-1
T / Temperature / 298.15 / K
Y / Yield of anodophilic / 17.1 / mg-M mg-S-1
YCH4 / Methane Yield / 0.3 / mL CH4 mg-S-1
qmax,a / Maximum anodophilc reaction rate / 5.32 / mg-S mg-x-1 d-1
qmax,m / Maximum methanogenic reaction rate / 8.2 / mg-S mg-x-1 d-1
µmax, a / Maximum anodophilic growth rate / 0.797 / d-1
µmax, m / Maximum methanogenic growth rate / 0.1 / d-1
Ks,a / Half-rate constant of anodophilics / 20 / mg-S L-1
Ks,m / Half-rate constant of methanogens / 80 / mg-S L-1
m / Electrons transfer per mol of mediator / 2 / mol-1 mol mediator -1
γ / Mediator molar mass / 663,400 / mg-M mol mediator-1
Mtotal / Mediator fraction / 0.05 / mg-M mg-x-1
KM / Mediator half rate constant / 0.01 / mg-M L-1
Kd,a / Decay rate of anodophilc microorganism / 0.02*µmax,a / d-1
Kd,m / Decay rate of methanogenic microorganism / 0.02*µmax,m / d-1
Xmax,a / Anodophilc biofilm space limitation / 512.5 / mg-x L-1
Xmax,m / Methanogenic biofilm space limitation / 525 / mg-x L-1
Kx / Steepness Factor / 0.04 / L mg-x-1
Rmin / Minimum internal resistance / 31 / Ω
Rmax / Maximum internal resistance / 2000 / Ω
Emin / Minimum open circuit potential / 0.1 / V
Emax / Maximum open circuit potential / 0.7 / V
KR / Steepness factor correction coefficient / 0.0818 / L mg-x-1

Table S2(b). Parameters of the MBER-2 for membrane and nutrients removal performances

Parameters / Description / Value / Units
η / Water apparent viscosity / 0.00089 / Pa S
A / Membrane surface area / 0.0261 / m-2
ρ / Specific cake resistance / 100000000 / m g-1
γ / Coefficient of β / 0.05
Y / Yield coefficient of the substrate consumption / 1.2 / mg-M mg-S-1
µs,max / Maximum growth rate / 0.12 / d-1
Jair / Air crossflow / 120 / m3 m-2 d-1
Ks / Half saturation of substrate / 10 / g m-3
Kair / Half saturation of airflow / 4.6*10-5 / g m-3
β / Resistance of detachable cake by air crossflow / 0.01 / m-1
Q / Inflow / 0.0019 / m3 d-1
V / Volume of cathodic compartment / 0.001 / m3
Dn / Diffusion coefficient for nitrate ions through AEM / 0.0293 / d-1
αn / Fraction of electrons for nitrate ions migration / 0.13

Table S3.Parameters of the MBER-3 for electrical and membrane performance.

Parameters / Description / Value / Units
F / Faraday constant / 96,485 / C mol-1
R / Ideal gas constant / 8.314 / J K-1 mol-1
T / Temperature / 298.15 / K
Y / Yield of anodophilic / 2.04 / mg-M mg-S-1
YCH4 / Methane Yield / 0.3 / mL CH4 mg-S-1
qmax,a / Maximum anodophilc reaction rate / 6.32 / mg-S mg-x-1 d-1
qmax,m / Maximum methanogenic reaction rate / 8.2 / mg-S mg-x-1 d-1
µmax, a / Maximum anodophilic growth rate / 0.797 / d-1
µmax, m / Maximum methanogenic growth rate / 0.1 / d-1
Ks,a / Half-rate constant of anodophilics / 20 / mg-S L-1
Ks,m / Half-rate constant of methanogens / 80 / mg-S L-1
m / Electrons transfer per mol of mediator / 2 / mol-1 mol mediator -1
γ / Mediator molar mass / 663,400 / mg-M mol mediator-1
Mtotal / Mediator fraction / 0.05 / mg-M mg-x-1
KM / Mediator half rate constant / 0.01 / mg-M L-1
Kd,a / Decay rate of anodophilc microorganism / 0.02*µmax,a / d-1
Kd,m / Decay rate of methanogenic microorganism / 0.02*µmax,m / d-1
Xmax,a / Anodophilc biofilm space limitation / 512.5 / mg-x L-1
Xmax,m / Methanogenic biofilm space limitation / 525 / mg-x L-1
Kx / Steepness Factor / 0.04 / L mg-x-1
Rmin / Minimum internal resistance / 31 / Ω
Rmax / Maximum internal resistance / 2000 / Ω
Emin / Minimum open circuit potential / 0.1 / V
Emax / Maximum open circuit potential / 0.7 / V
KR / steepness factor correction coefficient / 0.0818 / L mg-x-1
J / Water flux / 4.2 / L m-2 h-1
f / Constant in eq. 25 / 89330000
ULr / Hydraulic crossflow on membrane surface / 0.000634 / m s-1
c / Constant in eq. 25 / -1.1
d / Constant in eq. 25 / 0.376
e / Constant in eq. 25 / 0.532
X / Suspended solids concentration in the mixed liquor / 200 / mg L-1

S1