SENSITIVITY ANALYSIS OF A MATHEMATICAL
MODEL FOR MICROBIAL DESALINATION
CELLS
Merna Heshama a ; Abdelsalam Elawwadb b ; Ahmed Mekawy c and
Mohamed Hamdy Nour a
a Irrigation and Hydraulics Engineering Dept., Faculty of Engineering, Cairo
University, El-Gamaa St., 12613 Giza, Egypt.
b Environmental Engineering Dept., Faculty of Engineering, Cairo University, El-
Gamaa St., 12613 Giza, Egypt.
c Sanitary and Environmental Engineering Institute, Housing and Building National
Research Center, 87 Tahir St., 11511 Dokki, Giza, Egypt.
cE-mail – aabd_ekmaguid@yahoo.com
Key Words: Microbial Desalination cell – Mathematical model – Energy
–Desalination –Wastewater treatment
ABSTRACT
Microbial Desalination Cell (MDC) is new developed technology
that treats wastewater, generate electrical energy from it and desalinate
saline water. However, this complicated technology faces many
challenges to be implemented on a large scale that’s why mathematical
modelling for MDC is very essential. In this paper, A previous complex
mathematical model was studied. Sensitivity analysis was conducted for
model parameters to evaluate the influence of each parameter on the
dynamics of MDC including electric current, desalination and COD
removal to increase understanding of the relationships between input and
output parameters in the model. Based on sensitivity results, the top
effective parameters controlling the performance of MDC are maximum
anodophilic microorganisms growth rate, maximum substrate
consumption rate by anodophilic microorganisms and mediator yield.
INTRODUCTION
Bioelectrochemical systems (BES) are systems that their function
is wastewater treatment and energy recovery through converting
chemical energy embedded in wastewater to electrical energy through
microbial-electrochemical reactions [1]. Microbial desalination cell
(MDC) is one of bio electrochemical system types that desalinate salty
water using current generated from oxidation of organic matter so it
achieves three main goals: energy production, wastewater treatment and
desalination [2].MDCs concept of operation is similar to electrodialysis
desalination technology , however it uses electrical energy converted
from chemical energy in wastewater [3]. Since BES are complicated
Egypt. J. of Appl. Sci., 36 (7-8) 2021 110-122
systems , comprehensive mathematical models is important to understand
the dynamic relation between physical , chemical , biological and
electrochemical processes , so BES modelling is essential step towards
optimization and scaling up [4] . However , Few MDC models are
reported in the literature [5]. Ping Model is a MDC model with many
parameters that increases the complexity of the model. In this paper, A
sensitivity analysis was conducted for the parameters of Ping Model to
determine the effect of each parameters on the MDC performance. The
analysis was conducted using MATLAB and results of the most
important parameters are presented.
Materials and Methods (MDC Model[6])
1. Mass Balances for Substrate
The following equations shows the mass balance for the substrate
concentration in anode chamber
+Danode (Sin-S)
Danode=
Where t is time (d) ;S is substrate concentration (mg-S.L-1) ; Sin is
influent substrate concentration (mg-S.L-1); Ca and Cm are
concentrations for anodophilic and methanogenic microorganisms (mgx.
L-1); and are substrate consumption rate by anodophilic and
methanogenic microorganisms (mg-S .mg-x-1 .d-1); Danode is dilution
rate in anode chamber (day -1); Qin is the influent flow rate of substrate
(L.day-1); Vanode is the volume of anode chamber (L); and
are maximum substrate consumption rate by anodophilic and
methanogenic microorganisms (mg-S .mg-x-1 .d-1) ;K,a and K,m are the
half saturation constants for anodophilic and methanogenic
microorganisms (mg-S.L-1); Mox is oxidized mediator fraction per
anodophilic microorganisms (mg-M·mg-a−1); KM is half saturation
constant for mediators (mg-M.L-1).
Mass Balances for Microorganisms
The following equations shows the mass balance for microbial
population in anode chamber. The microbial population include
111 Egypt. J. of Appl. Sci., 36 (7-8) 2021
anodophilic microorganisms that release electrons from consumption of
organic matter and methanogenic microorganisms that convert substrate
into methane:
-
-
( ))
( ))
Where a and m are growth rate of anodophilic and methanogenic
microorganisms (d-1); Kd,a and Kd,m are decay rates of anodophilic and
methanogenic microorganisms; and are maximum growth
rate of anodophilic and methanogenic microorganisms ( d-1) ; αa and αm
are the dimensionless biofilm retention constants ; Ka,x and Km,x are
steepness factors of anodophilic and methanogenic microorganism
(L.mg-x-1); C a,max and C m,max are maximum attainable
concentration for anodophilic and methanogenic microorganisms (mgx.
L-1)
2. Mass balance for Mediators
The intracellular mediator exists either in its oxidized and reduced
form. The following equation shows the mass balance for oxidized
mediators
M total=M red+Mox
Where Mred and Mtotal are reduced and total mediator fraction per
anodophilic microorganisms (mg-M·mg-a−1); Y is the mediator yield
(mg-M .mg-S-1); IMDC is the MDC current (A) ; F is faraday constant
(A. d. mole-1); γ is mediator molar mass (mg-M molmed -1); ne is the
number of electrons transferred per mole of mediator (mole–e- .mole
Egypt. J. of Appl. Sci., 36 (7-8) 2021 112
med -1)
3. Mass Balance for Salt concentrations
The following equations shows the mass balance for the salt
concentration in the three chambers
( )
( )
( )-Danode (Csalt,a)
( )
Dsalt=
Csalt,a , Csalt,m, and Csalt,c are salt concentrations in anode , salt
and cathode chambers (mol-salt.L-1) ; C salt, in is influent salt
concentration (mol-salt.L-1) ;d is a membrane salt transfer coefficient
(day−1); Dsalt is dilution rate in middle chamber (day -1) ; Q salt is salt
flow rate (L.day−1); Vsalt is volume of salt chamber (L
4. Electrochemical equations
MDC current was determined by Ohm’s law:
)
Voc=Emin+ (Emax- Emin)
Rint=Rmin+ (Rmax- Rmin)
Where Voc is open circuit voltage (V); Emin and Emax are minimum
and maximum observed open circuit voltage (V); Rmin and Rmax are
minimum and maximum observed internal resistances (Ω); kr is the
constant that determines how fast the internal resistance respond to the
change in anodophilic microorganisms’ concentration (L·mg-a−1). R is
ideal gas constant (J.K-1.mole -1); R salt is resistance of salt solution (Ω)
; R membrane is mass transfer resistance through an exchange
membrane; Ranolyte is resistance of anolyte solution; Rext is the
external resistance
RESULTS AND DISCUSSION
1. Base case results
The model differential-algebraic equations were solved using
113 Egypt. J. of Appl. Sci., 36 (7-8) 2021
MATLAB and parameters values were determined from previous studies
[6], [7]. Substrate consumption increases by anodophilic and
methanogenic microorganisms with decreasing rate until it reaches
steady state after 8th day. Also, Current increases until it reaches the
steady state after 8th day so consequently salt removal from middle
chamber increases also with decreasing rate until it becomes constant
starting from 8th day.
Figure 1 Change of substrate concentration, current produced and
salt concentration in desalination chamber with time
2. Sensitivity analysis
The sensitivity analysis is conducted for MDC biological,
operating and design parameters to determine the effect of parameters on
the performance of MDC model. This analysis will use the local relative
sensitivity analysis method in which it will be carried out on all
parameters one by one by changing one of the parameters, while the
other parameters were fixed without any change. Local relative
sensitivity analysis [8] was used to determine the effect of changing the
parameter value as ratio between change in output value to change in
parameters value.
The following equation was used for each parameter:
( )
( )
Where the dependent time sensitivity for any parameter 'j' is Tj ; the
output value is P; the value of parameter j is xj ; the change in xj is xj;
Egypt. J. of Appl. Sci., 36 (7-8) 2021 114
the step of the change in this study is xj = 0.01xj.
3. Sensitivity analysis results
a) Maximum growth rate of microorganisms
The results of sensitivity analysis for the maximum growth rates
effect on current, substrate concentration and salt concentration in the
middle chamber are shown in Figure 2 . It is clear that anodophilic and
methanogenic microorganisms maximum growth rates (μa,max and
μm,max) are effectual parameters but methanogenic microorganisms
maximum growth rate is less effectual .The increase of anodophilic
microorganisms maximum growth rate cause higher rate of substrate
consumption so more decrease in substrate concentration until 7th day
then substrate concentration increases due biofilm space limitation .Also,
It cause increase in current produced and consequently the salt
concentration decreases until 7th day since anodophilic microorganisms
produce electrons from consumption of organic matter then after 7th day
current produced decreases and salt concentration increases. On the other
hand, the increase of methanogenic microorganisms’ maximum growth
rate cause decrease in substrate concentration and decrease in the current
produced as the methanogenic organisms don’t produce electrons from
substrate oxidation and consequently, salt concentration increases.
Figure 2 Relative sensitivity of current, substrate concentration and
salt concentration in the middle chamber with respect to a)
anodophilic maximum growth rates b) methanogenic
maximum growth rates
115 Egypt. J. of Appl. Sci., 36 (7-8) 2021
4. Half rate constant
The results of sensitivity analysis for the half rates constant effect
on current, substrate concentration and salt concentration in the middle
chamber are shown in Figure 3. It is clear that anodophilic
microorganisms, methanogenic microorganisms and mediators half rate
constant (Ka, Km and KM) are effectual parameters but mediators half
rate constant is the most effectual. As Ka increases, anodophilic
microorganisms growth rate and substrate consumption rate decrease
until 9th day. Due to decrease of anodophilic growth rate, the biofilm
space limitation effect is delayed so substrate concentration decrease and
current increases from 9th to 10th day. As Km increase, methanogenic
microorganisms’ growth rate decrease, so substrate concentration
increases, current decreases and salt concentration increases. As KM
increases, anodophilic microorganisms growth rate and substrate
consumption rate decrease, so substrate concentration increase and
current decrease that cause increase in salt concentration.
Figure 3 Relative sensitivity of current, substrate concentration and salt
concentration in the middle chamber with respect to half rate constant
of a) anodophilic microorganisms b) methanogenic microorganisms
and c) mediators
5. Decay rates
The results of sensitivity analysis for the decay rate effect on
current, substrate concentration and salt concentration in the middle
chamber are shown in Figure 4 . It is clear that the increase of
anodophilic and methanogenic microorganisms decay rate are ineffectual
parameters.
Egypt. J. of Appl. Sci., 36 (7-8) 2021 116
Figure 4 Relative sensitivity of current, substrate concentration and salt
concentration in the middle chamber with respect to decay rate of a)
anodophilic microorganisms and b) methanogenic microorganisms
6. Biofilm space limitation
The results of sensitivity analysis for the biofilm space limitation
effect on current, substrate concentration and salt concentration in the
middle chamber are shown in Figure 5.It is clear that the anodophilic and
methanogenic biofilm space limitation become more effectual from 7th
to 10th day because when space limitation value increases, it allows more
increase of anodophilic and methanogenic microorganisms concentration.
Figure 5 Relative sensitivity of current, substrate concentration and salt
concentration in the middle chamber with respect to a) anodophilic
biofilm space limitation and b) methanogenic biofilm space limitation
117 Egypt. J. of Appl. Sci., 36 (7-8) 2021
7.Maximum substrate consumption rate
The results of sensitivity analysis for the effect of maximum
substrate consumption rate on current, substrate concentration and salt
concentration in the middle chamber are shown in Figure 6. The increase
of methanogenic maximum substrate consumption rate has no effect on
current production or desalination but it increases the rate of substrate
oxidation by methanogenic organisms. However, the increase of
maximum substrate consumption rate by anodophilic microorganisms
cause sharp decrease in substrate concentration. Also, It cause increase in
current and consequently decrease in salt concentration as increase of
maximum substrate consumption rate reduce oxidized mediators
concentration so it cause a decrease in concentration losses that leads to
increase in current produced.
Figure 6 Relative sensitivity of current, substrate concentration and salt
concentration in the middle chamber with respect to maximum
substrate consumption rate by a) methanogenic microorganisms and b)
anodophilic microorganisms
8.Mediator yield
The results of sensitivity analysis for the effect of mediator yield
on current, substrate concentration and salt concentration in the middle
chamber are shown in Figure 7. Concerning the current and salt
concentration in middle chamber, it is clear that mediator yield is very
effectual parameter as it causes more current production and salt removal
but it has minor effect on substrate concentration.
Egypt. J. of Appl. Sci., 36 (7-8) 2021 118
Figure 7 Relative sensitivity of current, substrate concentration and salt
concentration in the middle chamber with respect to mediator yield
9.Diffusion coefficient
The results of sensitivity analysis for the effect of diffusion
coefficient on current, substrate concentration and salt concentration in
the middle chamber are shown in Figure 8. It is clear that increase of
diffusion coefficient is very effectual only in increasing salt removal in
the middle chamber, however it has negligible effect on current.
Figure 8 Relative sensitivity of current, substrate concentration and salt
concentration in the middle chamber with respect to diffusion
coefficient
119 Egypt. J. of Appl. Sci., 36 (7-8) 2021
10.Number of electrons transferred per mole of mediator
The results of sensitivity analysis for the effect of diffusion
coefficient on current, substrate concentration and salt concentration in
the middle chamber are shown in Figure 8 . It is very clear that number
of electrons transferred per mediator is a very effectual parameter.
Figure 9 Relative sensitivity of current, substrate concentration and salt
concentration in the middle chamber with respect to number of
electrons transferred per mole of mediator
CONCLUSION
In this study, it is very clear that MDC model is very complicated
and multivariable system that requires more analysis to be easily
understood and used. Sensitivity analysis is one of the important tools
that help to understand this system . Sensitivity analysis that was
conducted for the MDC model shows the effect of each parameter on the
performance of MDC inculding COD removal , desalination and
electrical current produced . Top parameters that affect MDC
performance are maximum substrate consumption rate by anodophilic
microorganis, the maximum growth rate of these microorganisms and
mediator yield so model users have to focus on reestimation of these
parameters’ values for better prediction of results.
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REFERENCES
[1] Yang, E. ; K.J. Chae ; M.J. Choi ; Z. He and I.S. Kim(2019):
“Critical review of bioelectrochemical systems integrated with
membrane-based technologies for desalination, energy selfsufficiency,
and high-efficiency water and wastewater treatment,”
Desalination, 452: 40–67.
[2] Elawwad, A. ; D.Z. Husein ; M. Ragab and A. Hamdy(2020):
“Enhancing the performance of microbial desalination cells using
δMnO2/graphene nanocomposite as a cathode catalyst,” J. Water
Reuse Desalin., 10 (3):214–226.
[3] Xu, C. and et al., (2020): “Simultaneous bioelectricity generation,
desalination, organics degradation, and nitrogen removal in air–
cathode microbial desalination cells,” SN Appl. Sci., 2(2):1–11.
[4] Gadkari, S. ; S. Gu and J. Sadhukhan(2018): “Towards
automated design of bioelectrochemical systems: A comprehensive
review of mathematical models,” Chem. Eng. J., 343: 303–316.
[5] Rahman, S. and et al.,(2021): “Towards upscaling microbial
desalination cell technology: A comprehensive review on current
challenges and future prospects,” J. Clean. Prod., 288: 125597.
[6] Ping, Q. ; C. Zhang ; X. Chen ; B. Zhang ; Z. Huang and Z.
He(2014): “Mathematical model of dynamic behavior of microbial
desalination cells for simultaneous wastewater treatment and water
desalination,” Environ. Sci. Technol., 48: 13010–13019.
[7] Pinto, R.P. ; B. Srinivasan ; M.F. Manuel and B. Tartakovsky
(2010): “A two-population bio-electrochemical model of a
microbial fuel cell,” Bioresour. Technol., 101(14): 5256–5265 .
[8] Zeng, Y. ; Y.F. Choo ; B.H. Kim and P. Wu (2009): “Modelling
and simulation of two-chamber microbial fuel cell,” J. Power
Sources, 195(1):79–89.
تحميل الحساسية لنموذج رياضي لخلايا تحمية المياه الميکروبية
ميرنا هشام أ ، عبد السلام العواد ج ، أحمد محمد عبد المجيد مکاوي ب ، محمد حمدي نورأ
أ– قسم ىندسة الرى والييدروليکا - کمية اليندسة - جامعة القاىرة
ب – قسم اليندسة البيئية - کمية اليندسة - جامعة القاىرة
ج – معيد اليندسة الصحية والبيئية ، المرکز القومي لبحوث الإسکان والبناء
نبذة مختصرة
التي تحول المرکبات )MFCs( من تقنيات المعالجة الجديدة ىي خلايا الوقود الميکروبية
العضوية في مياه الصرف الصحي إلى طاقة کيربائية من خلال سمسمة من العمميات الفيزيائية
تعد من التقنية )MDC( والکيميائية والبيولوجية والکيروکيميائية. خمية التحمية الميکروبية
الجديدة والمطورة لمعالجة مياه الصرف الصحي ، وتوليد الطاقة الکيربائية منيا وتحمية المياه
المالحة. تواجو ىذه التکنولوجيا الجديدة العديد من التحديات لامکانية تنفيذىا عمى نطاق واسع
121 Egypt. J. of Appl. Sci., 36 (7-8) 2021
تعتير ضرورية لمغاية. MDC وليذا السبب في أن معرفة کاممة لمنمذجة الرياضية ليذه التقنية
و تم د ا رسة تأثير کل متغير عمى Matlab باستخدام MDC في ىذا البحث تم د ا رسة نموذج
ثم تم استخدام نتائج التحميل لتبسيط المعادلات من خلال استبعاد المتغي ا رت غير MDC أداء
الفعالة. کما تم التحقق من صحة النموذج المخفض لتدفق الدفعات الدورية باستخدام نتائج
اختبا ا رت معممية تم الحصول عمييا من الم ا رجع المنشورة ثم تم مقارنتو مع نموذج اخر باستخدام
ثم تم MDC نفس النتائج المعممية. تمت د ا رسة تأثير متغي ا رت التشغيل المختمفة عمى أداء نظام
إضافة بعض التعديلات عمى معادلات النموذج لتتوافق مع الاتجاىات الملاحظة في الد ا رسات
لأىداف MDCs التجريبية. يتم تقديم النموذج النيائي و الذي يمکن استخدامو في تصميم ىذه
معالجة مختمفة
الکممات الدالة: معالجة مياه الصرف الصحي - تحمية المياه،الطاقة - خلايا الوقود الميکروبية
- خلايا التحمية الميکروبية
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