- Email: [email protected]
Transport equations in an enzymatic glucose fuel cell
Accepted Manuscript Research paper Transport equations in an enzymatic glucose fuel cell Soham Jariwala, Balaji Krishnamurthy PII: DOI: Reference:
S0009-2614(17)31072-2 https://doi.org/10.1016/j.cplett.2017.11.055 CPLETT 35269
To appear in:
Chemical Physics Letters
Received Date: Accepted Date:
19 July 2017 27 November 2017
Please cite this article as: S. Jariwala, B. Krishnamurthy, Transport equations in an enzymatic glucose fuel cell, Chemical Physics Letters (2017), doi: https://doi.org/10.1016/j.cplett.2017.11.055
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Transport equations in an enzymatic glucose fuel cell by Soham Jariwala and Balaji Krishnamurthy*, Department of Chemical Engineering, BITS Pilani, Hyderabad 500078, India.
Abstract: A mathematical model is developed to study the effects of convective flux and operating temperature on the performance of an enzymatic glucose fuel cell with a membrane. The model assumes isothermal operating conditions and constant feed rate of glucose. The glucose fuel cell domain is divided into five sections, with governing equations describing transport characteristics in each region, namely - anode diffusion layer, anode catalyst layer (enzyme layer), membrane, cathode catalyst layer and cathode diffusion layer. The mass transport is assumed to be one-dimensional and the governing equations are solved numerically. The effects flow rate of glucose feed on the performance of the fuel cell are studied as it contributes significantly to the convective flux. The effects of operating temperature on the performance of a glucose fuel cell are also modeled. The cell performances are compared using cell polarization curves, which were found compliant with experimental observations.
Keywords: glucose, convection, fuel cell, hydrogen, diffusion, temperature.
*Corresponding author; E mail: [email protected]
1
1. Introduction Several researchers have studied the performance of glucose fuel cells over the last decade1-8. Glucose fuel cells have been studies for applications in biological implants and pacemakers. Glucose fuel cells are of two types-microbial and enzymatic. The anodic reaction in an enzymatic glucose fuel cell is given by the following reaction3, taking place in basic medium
C H O + 2OH → C H O + H O + 2e
(1)
with an anode potential of -0.853 V/SHE (Standard Hydrogen electrode). The cathodic reaction is given by the reaction
1 (2) O + H + 2e → H O 2 with a cathode potential of 0.403 vs SHE. The overall reaction of the cell is given by the relation3
C H O + O → C H O , = 1.256
(3)
Figure 1 shows the schematic of enzymatic glucose fuel cell with ion exchange membrane. The membrane used in enzymatic glucose fuel cells is a cation exchange membrane (Nafion), which allows transport of proton thorough it, while acting as a barrier for glucose molecules. Very few mathematical models are available in literature to study the performance of an enzymatic glucose fuel cell. Vladimir Rubin1-2 has studied the steady state performance of an enzymatic glucose fuel cells. Basu6 had studied the theoretical performance of an anion exchange based Direct glucose fuel cell. Der Sheng Chan7 has modeled the anode of an enzyme based bio fuel cell. Yin Song8 has modeled the performance of enzymatic bio fuel cell with three dimensional microelectrodes. Basu9 has studied the effect of overpotentials on the performance of direct glucose alkaline fuel cell. Annepu10 has modeled the performance of a five-layer direct glucose fuel cell. In this paper, we attempt at modeling the effects of convective flux and temperature on the performance of an enzymatic glucose fuel cell, using cation exchange membrane and glucose oxidase enzyme (GO ), which has not been studied in published literature so far, to our knowledge. 2. Model development 2.1 Assumptions 2
a) The model assumes that the glucose and hydrogen ion transport through the cell occurs as a plug flow, with total lateral mixing, reducing the domain to one dimensional. b) The glucose fuel cell domain is assumed to be divided into 5 sections, namely - the anode diffusion layer (ADL), the anode catalyst layer or enzyme layer (EL), the anion exchange membrane, the cathode catalyst layer (CCL) and the cathode diffusion layer (CDL). The schematic of the fuel cell is shown in Figure 1 with the dimensions (taken from literature). c) It is assumed that the membrane is impermeable to glucose but permeable to hydrogen ions. Thus, the concentration of the glucose is modeled only till the membrane. d) The reaction of the glucose to form hydrogen ions is assumed to occur only in the enzyme layer, in presence of the enzyme, glucose oxidase (GOx). e) The unreacted glucose and the hydrogen ions diffuse across the cell to the cathode. The unreacted glucose is assumed not to react at the cathode. f) The model assumes that the transport of glucose is solely by diffusion and convection. The effect of the potential drop across the glucose fuel cell on glucose molecules is considered to be negligible (glucose is not a charged species). g) The potential drop across the membrane is considered to a significant factor in the transport of hydrogen ions across the membrane. The effect of potential drop on the transport of hydrogen ions across the enzyme layer and the cathode catalyst layer is considered negligible given the thickness of these layers (Schematic 1). Diffusion and convection are considered to be the major factors in the transport of hydrogen ions across these catalyst layers. h) The effect of potential on the reaction kinetics at the anode and cathode is neglected. i) The cathode diffusion layer is assumed to be a reaction zone, where hydrogen ions react with oxygen molecules. j) The flow domain is assumed to be dilute, such that Fick’s law of diffusion holds. 2.2 Model Equations Figure 1 shows the schematic of the glucose fuel cell being modeled, illustrating the flow domain and various transport regions. The mass transport equations in glucose fuel cell can be 3
obtained from general component mass balance for a reaction system. The system is assumed to be dilute; hence Fick’s law of diffusion holds. The system of linear partial differential equations would be solved for the concentrations of glucose and hydrogen ions in the fuel cell domain. The general transient mass balance11 can be stated as, .
= " ∇ − ∇( %) + ∇(& ' ( ∇Φ) + * + , !
(4)
-/
Since the glucose molecules carry no charge, there is no migration term arising due to the presence of potential gradient. The transport and consumption of glucose molecules is dictated by diffusion, convection as well as oxidation in glucose oxidase enzyme layer. Migration will come into effect when describing the concentration of hydrogen ions as hydrogen ions are positively charged and will be repelled by the presence of potential gradient across the region of membrane electrode assembly. Figure 2 shows the schematic according to which the enzymatic glucose fuel cell is modeled. The mass transport equations across each of the regions is given below. 2.21 Anode Diffusion layer(ADL) (0 ≤ 2 < 4567 ) The transport equations for glucose in the anode diffusion layer is given by11
89 :;< =9 89 :;< =9 89 :;< =9 = "89 :;< =9 ,567 −% ! 2 2 The transport equation for hydrogen ions in the anode diffusion layer is given by11 : > : > : > > = ": ,567 − % ! 2 2 2.22 Enzyme layer (EL) (4567 ≤ 2 < 4567 + 4?7)
(5)
(6)
Glucose reacts with glucose oxidase enzyme in this layer, resulting in depletion of glucose and generation of hydrogen ions. The reaction is assumed to follow Michaelis-Menten model1-2, the rate expression being,
+@ =
ABC DEF DEF + GHH
4
(7)
Here, ABC is the maximum rate of reaction for given concentration of enzyme and DEF is the
concentration of substrate and ABC = GBI DF. KMM is the Michaelis Menten constant. Since the substrate is the glucose molecule, the glucose transport equation in enzyme layer becomes
89 :;< =9 89 :;< =9 89 :;< =9 GBI D F 89 :;< =9 = "89 :;< =9 ,?7 − % − ! 2 2 GHH + 89 :;< =9
(8)
where the last term in the equation represents the glucose consumption term. Similarly, the transport equation for hydrogen ions in the enzyme layer can be written as J8K> JI
= ": >,?7
J <8K> JL <
−%
J8K> JL
NOPQ D?F8R9 K;< S9
+2·
(9)
NTT 8R9K;< S9
where the last term in the equation represents the hydrogen ion generation term. 2.23 Membrane (4567 + 4?7 ≤ 2 < 4567 + 4?7 + 4HA ) The transport equation (diffusion and convection) for glucose in the membrane is given by J8R9 K;< S9 JI
= "89 :;< =9 ,HA
J < 8R9 K;< S9 JL <
−%
J8R9 K;< S9
(10)
JL
The transport equation for the transport of hydrogen ions through the membrane is given by J8K> JI
= ": >,HA
J <8K> JL <
−%
J8K> JL
UV ∗ J8K>
− ": > 7
TXY
(11)
JL
The first term represents the diffusion term, the second term represents the convection term and the last term represents the migration term. The term ∆ ∗ represents the dimensionless potential given by the term12
[∆V \]
.
2.24 Cathode Catalyst Layer (CCL) and Cathode Diffusion layer (CDL) The hydrogen ions react at the cathode catalyst region, combining with oxygen molecules delivered through the cathode compartment. The product formed is water and free electrons, which are transported through the cathode upon completion of circuit. The transport equation for the cathode catalyst layer ( 4( − 4 "4 − 4 4 ≤ 2 < 4( − 4 "4 ) is given by J8K> JI
= ": > ,887
J <8K> JL <
−%
J8K> JL
;
− G,887 D^ F< : > ,
(12)
Similarly, for cathode diffusion layer (4[8 − 4867 ≤ 2 < 4[8 ) which is also a reaction
5
zone, the equation for hydrogen ion transport becomes (it is to be noted that the diffusivity for the cathode catalyst layer is different from the diffusivity of the cathode diffusion layer) : > : > : > = ": >,867 −% − G,867 D^ F : > , ! 2 2
(13)
where the last term denotes the reaction in the cathode diffusion layer. 2.25 Initial and Boundary conditions The concentration of protons and glucose in the fuel cell system follows a system of linear partial differential equation. Hence, both initial and boundary conditions are required to obtain a solution. The glucose concentration in the cell domain is initially assumed to be 0 _, with glucose entering from the left of the anode compartment, with a constant feed concentration at the beginning of computation time. Therefore, At ! = 0,
89 :;< =9 = 0 and : > = 0, ∀ 2
At ! > 0
bc
89 :;< =9 = 89 :;< =9 and : > = 0, at 2 = 0
(14)
(15)
Also, the concentrations at interfaces of all layer are assumed to be in equilibrium. Therefore,
d 8
9 :;< =9
e
L/Lf gL
= d 89 :;< =9 eL/L gL
(16)
= d : > |L/Lf gL
(17)
where 2 is the coordinate of each interface. Similarly, At 2 = 4[8 ,
d
: > |L/Lf gL
f
i : > = 0, ∀ ! i2
(18)
2.26 Operating Temperature The behavior of fuel cell is strongly affected by the change in operating temperature, as the rate constants and diffusivities are functions of temperature. The rate constants are assumed to vary as per Arrhenius equations and diffusivity is assumed obey Eyring’s equation. ?l
" = "j k ℜ] 6
(19)
n = nj k
? P ℜ]
(20)
2.27 Current Density The current density is calculated at the cathode interface. The current density is a function of hydrogen ion gradient, and can be written as,
i : > p i2 L/LOPQqrls being the coordinate of the cathode plate. o = −( · ": > d
2BItuc
(21)
2.3 Solution methodology For obtaining concentrations of hydrogen ions and glucose, a system of partial differential equations (PDEs) has to be solved. The resulting system is linear, but cannot be solved analytically without complex mathematical treatment. Therefore, it was solved numerically, by reducing the system of PDEs to a system of ordinary differential equations by employing finite difference methods for reducing derivative terms by discretizing the domain into finite nodes. The concentrations were then defined for these discrete nodes. Each layer was divided into 20 nodes, two equations being solved at each node, one for concentration hydrogen ions and other for concentration of glucose molecule. Commercial solver ‘ode15s’ which comes bundled with MATLAB was employed to solve the system. The cathodic current density was calculated using second order finite difference to approximate the derivative of concentration of hydrogen ions. 3.0 Results and Discussion Figure 3 shows the variation of the concentration of glucose with time across the anode diffusion layer, the enzyme layer and the membrane as a function of inlet glucose concentration and inlet flow rate. It is to be noted that the first layer in the figure denotes the anode diffusion layer, followed by the enzyme layer, the membrane, the cathode catalyst layer and the cathode diffusion layer. At initial values of time, the diffusional limitations in the anode diffusion layer causes a steep drop in the concentration profile of the glucose across the ADL. However, with increasing time, the concentration profile across the ADL increases and levels off. The reaction in the enzyme layer is indicated by the steep drop in concentration of the glucose across the enzyme layer. The parameter values listed in Table 1 were used for simulations presented in Figure 3 to 12.
7
The transient variation of hydrogen ions can be seen in Figure 4 across the anode diffusion layer, the enzyme layer, the membrane, the cathode diffusion layer and the cathode catalyst layer. Complete consumption of hydrogen ion is assumed in the cathode catalyst layer and cathode diffusion layer. It can be seen that the concentration of the hydrogen ions is seen to increase in the enzyme layer where the glucose oxidation reaction happens but is seen to decrease after that due to consumption of the hydrogen ions in the cathode catalyst layer. Due to the presence of potential across the cell, the hydrogen ions experience a backward diffusive flux, which is reflected in Figure 4. Figure 5 shows the variation of glucose concentration across the anode diffusion layer, the enzyme layer and the membrane as a function of inlet glucose flow rate. There is no drop in the concentration of glucose species across the anode diffusion layer since convective flux due to high flow rates overcome the low diffusive fluxes.
There is a steep drop in the glucose
concentration in the enzyme layer, due to reaction with glucose oxidase. Figure 6 shows the variation of glucose concentration across the ADL, EL and the membrane as a function of temperature. Since the diffusion coefficients increase with increase in temperature, there is a notable increase in diffusive flux with temperature in the ADL. The concentration of the glucose is seen to drop steeply across the enzyme layer indicating that the reaction rate is a strong function of temperature. Operating temperature also plays a major role in determining consumption rate of glucose molecules, evident by steeper drop in the concentration of glucose in enzyme layer. Figure 7 shows the variation of hydrogen ions across the ADL, EL, membrane and the CCL as a function of temperature. The trend in hydrogen ion concentration can be deduced from Figure 6, as higher operating temperature results in higher concentration of hydrogen ions in the enzyme layer. Figure 8 shows the concentration of hydrogen ions across the enzyme layer as a function of the thickness of the enzyme layer. It is seen that the higher the thickness of the enzyme layer, the higher the concentration of the hydrogen ions. More thickness of enzyme layer provides larger region and more sites for oxidation, resulting in increased generation of hydrogen ions. Figure 9 shows the polarization curve of the enzymatic glucose fuel cell as a function of time. The system achieves steady state in about 2 minutes, for given set of parameters. Figures 10 and 11 shows 8
the polarization curve of the enzymatic glucose fuel cell as a function of flow rate and temperature at steady state. The inlet concentration of glucose molecules strongly affects the polarization curves and cell performance. Figure 12 shows the polarization curve of the enzymatic glucose fuel cell as a function of the thickness of the enzyme layer. The performance of the fuel cell is seen to increase with increasing thickness of the enzyme layer. Figure 13 shows the comparison of modeling results with experimental data13. Simulation results are compared with experimental polarization curves obtained at 310 K and flow rate of 167 mm3/s. The results obtained from the model were found to compare well with the experimental data. Table 1 enlists the parameters used in carrying out simulations presented in Figure 13, with aforementioned temperature and flow rate values.
4. Conclusion A mathematical model is developed to study the concentration profiles of glucose and hydrogen ions across the enzymatic glucose fuel cell. The concentration profiles of glucose are studied across the anode diffusion layer, enzyme layer and the membrane. The concentration profiles of glucose are studies across the anode diffusion layer, enzyme layer, membrane and the cathode catalyst layer. The concentration profiles are found to be sensitive to temperature and inlet flow rate. The thickness of the enzyme layer is found to play a significant role in determining the concentration of hydrogen ions across the layer and in the overall performance of the cell. Model results are compared with experimental data and found to agree very well.
9
5.Figure Captions Figure 1 - Schematic diagram of a enzymatic glucose fuel cell representing various layers and their thicknesses. Figure 2 – Schematic diagram showing the modeling configuration of a enzymatic glucose fuel cell. Figure 3 - Variation of glucose concentration as a function of time in the fuel cell domain Figure 4 - Variation of concentration of hydrogen ions as a function of time in the fuel cell domain Figure 5 - Glucose concentration across the cell with varying flow rates. Figure 6 - Effect of temperature on glucose concentration profile across the cell. Figure 7 - Hydrogen ion concentration across the cell with varying operating temperatures. Figure 8 - Hydrogen ion concentration inside the enzyme layer as a function of the thickness of the enzyme layer. Figure 9 –Transient polarization curves Figure 10 – Polorization curves of the fuel cell with varying glucose feed flow rates. Figure 11 – Polorization curves of the fuel cell with various operating temperatures. Figure 12 – Polorization curves of the fuel cell as a function of thickness of enzyme layer Figure 13 - Comparison of polarization curves with experimental data.
10
6.Acknowledgements The authors wish to acknowledge BITS Pilani for enabling us to submit this article. 7. References 1.Vladmir R, Lea M, Physical modeling of enzymatic glucose fuel cells,
Advances in Chemical
Engineering and Science, 2013, 3, 218-224 2.Vladmir R, Current Voltage modeling of enzymatic glucose fuel cells, Advances in Chemical Engineering and Science, 2015, 5,164-170. 3.Gymama S , Tanmay Kulkarni, Enzymatic glucose fuel cell and its application, Biochip Tissue Chip, 2015, 5, 1-6. 4.Atanassov P, Chris A and Shelley M, Enzymatic bio fuel cells, Electrochemical Society Interface, 2007, 28-32. 5.Peter C, Donal Leech, Fully enzymatic glucose oxygen fuel cell, Anal. Chem, 2016,88, 2156-2163. 6.Ranoo P and Basu.S, Mathematical modeling and experimental verification of direct glucose anion exchange membrane fuel cell, Electrochimica Acta, (2013) 113 : 42– 53.
7.Sheng C, Der.J , Shing W, Dynamic Modeling of Anode Function in Enzyme-Based Biofuel Cells Using High Mediator Concentration Energies 2012, 5, 2524-2544;
8.Yin S, Varun P, Chunlei W, Modeling and Simulation of Enzymatic Biofuel Cells with Three-Dimensional Microelectrodes, Energies, 2014, 7, 4694-4709. 9.Basu D, Basu S, Mathematical modeling of overpotentials of direct glucose alkaline fuel cell and experimental validation, J. Solid State Electrochem, 2013, 17, 2927-2938. 10.J.J.Aneppu, Masters thesis, Mathematical simulation of direct glucose fuel cells, Governors State University, Fall 2015. 11. John Newman and K.E.Thomas Alyea, Electrochemical Systems, Wiley, 2004. 12. S.Phul, A.Despande and Balaji Krishnamurthy, The effect of potential drop on the capacity fading in lithium ion batteries, Electrochimica Acta, 164 (2015), 281-287.
11
8. Nomenclature
v ( w"4 4 _kx 4 "4 ℜ bc ! 2 " ,A % & ( ' ∇Φ + ,z o 2{ GBI G D^ F DF c B 4A 4H?5 "j nj
absolute temperature fuel cell anode diffusion layer enzyme layer/anode catalyst layer membrane cathode catalyst layer cathode diffusion layer universal gas constant concentration of component ‘y’ inlet feed concentration of component ‘y’ time co-ordinate on axis along length of the fuel cell diffusion coefficient of component ‘y’ in medium ‘x’. velocity or volume flux vector charge on species of component ‘y’ Faraday’s constant mobility of species ‘y’ electric field rate of formation of component ‘y’ in reaction ‘o’ total number of components current density co-ordinate of interface between two layers enzyme rate constant rate constant for cathode reaction concentration of oxygen gas concentration of glucose oxidase enzyme activation energy for diffusion per mole activation energy for reaction per mole length of layer ‘x’ length of membrane electrode assembly maximal diffusion constant maximal reaction rate constant
12
13
S0009-2614(17)31072-2 https://doi.org/10.1016/j.cplett.2017.11.055 CPLETT 35269
To appear in:
Chemical Physics Letters
Received Date: Accepted Date:
19 July 2017 27 November 2017
Please cite this article as: S. Jariwala, B. Krishnamurthy, Transport equations in an enzymatic glucose fuel cell, Chemical Physics Letters (2017), doi: https://doi.org/10.1016/j.cplett.2017.11.055
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Transport equations in an enzymatic glucose fuel cell by Soham Jariwala and Balaji Krishnamurthy*, Department of Chemical Engineering, BITS Pilani, Hyderabad 500078, India.
Abstract: A mathematical model is developed to study the effects of convective flux and operating temperature on the performance of an enzymatic glucose fuel cell with a membrane. The model assumes isothermal operating conditions and constant feed rate of glucose. The glucose fuel cell domain is divided into five sections, with governing equations describing transport characteristics in each region, namely - anode diffusion layer, anode catalyst layer (enzyme layer), membrane, cathode catalyst layer and cathode diffusion layer. The mass transport is assumed to be one-dimensional and the governing equations are solved numerically. The effects flow rate of glucose feed on the performance of the fuel cell are studied as it contributes significantly to the convective flux. The effects of operating temperature on the performance of a glucose fuel cell are also modeled. The cell performances are compared using cell polarization curves, which were found compliant with experimental observations.
Keywords: glucose, convection, fuel cell, hydrogen, diffusion, temperature.
*Corresponding author; E mail: [email protected]
1
1. Introduction Several researchers have studied the performance of glucose fuel cells over the last decade1-8. Glucose fuel cells have been studies for applications in biological implants and pacemakers. Glucose fuel cells are of two types-microbial and enzymatic. The anodic reaction in an enzymatic glucose fuel cell is given by the following reaction3, taking place in basic medium
C H O + 2OH → C H O + H O + 2e
(1)
with an anode potential of -0.853 V/SHE (Standard Hydrogen electrode). The cathodic reaction is given by the reaction
1 (2) O + H + 2e → H O 2 with a cathode potential of 0.403 vs SHE. The overall reaction of the cell is given by the relation3
C H O + O → C H O , = 1.256
(3)
Figure 1 shows the schematic of enzymatic glucose fuel cell with ion exchange membrane. The membrane used in enzymatic glucose fuel cells is a cation exchange membrane (Nafion), which allows transport of proton thorough it, while acting as a barrier for glucose molecules. Very few mathematical models are available in literature to study the performance of an enzymatic glucose fuel cell. Vladimir Rubin1-2 has studied the steady state performance of an enzymatic glucose fuel cells. Basu6 had studied the theoretical performance of an anion exchange based Direct glucose fuel cell. Der Sheng Chan7 has modeled the anode of an enzyme based bio fuel cell. Yin Song8 has modeled the performance of enzymatic bio fuel cell with three dimensional microelectrodes. Basu9 has studied the effect of overpotentials on the performance of direct glucose alkaline fuel cell. Annepu10 has modeled the performance of a five-layer direct glucose fuel cell. In this paper, we attempt at modeling the effects of convective flux and temperature on the performance of an enzymatic glucose fuel cell, using cation exchange membrane and glucose oxidase enzyme (GO ), which has not been studied in published literature so far, to our knowledge. 2. Model development 2.1 Assumptions 2
a) The model assumes that the glucose and hydrogen ion transport through the cell occurs as a plug flow, with total lateral mixing, reducing the domain to one dimensional. b) The glucose fuel cell domain is assumed to be divided into 5 sections, namely - the anode diffusion layer (ADL), the anode catalyst layer or enzyme layer (EL), the anion exchange membrane, the cathode catalyst layer (CCL) and the cathode diffusion layer (CDL). The schematic of the fuel cell is shown in Figure 1 with the dimensions (taken from literature). c) It is assumed that the membrane is impermeable to glucose but permeable to hydrogen ions. Thus, the concentration of the glucose is modeled only till the membrane. d) The reaction of the glucose to form hydrogen ions is assumed to occur only in the enzyme layer, in presence of the enzyme, glucose oxidase (GOx). e) The unreacted glucose and the hydrogen ions diffuse across the cell to the cathode. The unreacted glucose is assumed not to react at the cathode. f) The model assumes that the transport of glucose is solely by diffusion and convection. The effect of the potential drop across the glucose fuel cell on glucose molecules is considered to be negligible (glucose is not a charged species). g) The potential drop across the membrane is considered to a significant factor in the transport of hydrogen ions across the membrane. The effect of potential drop on the transport of hydrogen ions across the enzyme layer and the cathode catalyst layer is considered negligible given the thickness of these layers (Schematic 1). Diffusion and convection are considered to be the major factors in the transport of hydrogen ions across these catalyst layers. h) The effect of potential on the reaction kinetics at the anode and cathode is neglected. i) The cathode diffusion layer is assumed to be a reaction zone, where hydrogen ions react with oxygen molecules. j) The flow domain is assumed to be dilute, such that Fick’s law of diffusion holds. 2.2 Model Equations Figure 1 shows the schematic of the glucose fuel cell being modeled, illustrating the flow domain and various transport regions. The mass transport equations in glucose fuel cell can be 3
obtained from general component mass balance for a reaction system. The system is assumed to be dilute; hence Fick’s law of diffusion holds. The system of linear partial differential equations would be solved for the concentrations of glucose and hydrogen ions in the fuel cell domain. The general transient mass balance11 can be stated as, .
= " ∇ − ∇( %) + ∇(& ' ( ∇Φ) + * + , !
(4)
-/
Since the glucose molecules carry no charge, there is no migration term arising due to the presence of potential gradient. The transport and consumption of glucose molecules is dictated by diffusion, convection as well as oxidation in glucose oxidase enzyme layer. Migration will come into effect when describing the concentration of hydrogen ions as hydrogen ions are positively charged and will be repelled by the presence of potential gradient across the region of membrane electrode assembly. Figure 2 shows the schematic according to which the enzymatic glucose fuel cell is modeled. The mass transport equations across each of the regions is given below. 2.21 Anode Diffusion layer(ADL) (0 ≤ 2 < 4567 ) The transport equations for glucose in the anode diffusion layer is given by11
89 :;< =9 89 :;< =9 89 :;< =9 = "89 :;< =9 ,567 −% ! 2 2 The transport equation for hydrogen ions in the anode diffusion layer is given by11 : > : > : > > = ": ,567 − % ! 2 2 2.22 Enzyme layer (EL) (4567 ≤ 2 < 4567 + 4?7)
(5)
(6)
Glucose reacts with glucose oxidase enzyme in this layer, resulting in depletion of glucose and generation of hydrogen ions. The reaction is assumed to follow Michaelis-Menten model1-2, the rate expression being,
+@ =
ABC DEF DEF + GHH
4
(7)
Here, ABC is the maximum rate of reaction for given concentration of enzyme and DEF is the
concentration of substrate and ABC = GBI DF. KMM is the Michaelis Menten constant. Since the substrate is the glucose molecule, the glucose transport equation in enzyme layer becomes
89 :;< =9 89 :;< =9 89 :;< =9 GBI D F 89 :;< =9 = "89 :;< =9 ,?7 − % − ! 2 2 GHH + 89 :;< =9
(8)
where the last term in the equation represents the glucose consumption term. Similarly, the transport equation for hydrogen ions in the enzyme layer can be written as J8K> JI
= ": >,?7
J <8K> JL <
−%
J8K> JL
NOPQ D?F8R9 K;< S9
+2·
(9)
NTT 8R9K;< S9
where the last term in the equation represents the hydrogen ion generation term. 2.23 Membrane (4567 + 4?7 ≤ 2 < 4567 + 4?7 + 4HA ) The transport equation (diffusion and convection) for glucose in the membrane is given by J8R9 K;< S9 JI
= "89 :;< =9 ,HA
J < 8R9 K;< S9 JL <
−%
J8R9 K;< S9
(10)
JL
The transport equation for the transport of hydrogen ions through the membrane is given by J8K> JI
= ": >,HA
J <8K> JL <
−%
J8K> JL
UV ∗ J8K>
− ": > 7
TXY
(11)
JL
The first term represents the diffusion term, the second term represents the convection term and the last term represents the migration term. The term ∆ ∗ represents the dimensionless potential given by the term12
[∆V \]
.
2.24 Cathode Catalyst Layer (CCL) and Cathode Diffusion layer (CDL) The hydrogen ions react at the cathode catalyst region, combining with oxygen molecules delivered through the cathode compartment. The product formed is water and free electrons, which are transported through the cathode upon completion of circuit. The transport equation for the cathode catalyst layer ( 4( − 4 "4 − 4 4 ≤ 2 < 4( − 4 "4 ) is given by J8K> JI
= ": > ,887
J <8K> JL <
−%
J8K> JL
;
− G,887 D^ F< : > ,
(12)
Similarly, for cathode diffusion layer (4[8 − 4867 ≤ 2 < 4[8 ) which is also a reaction
5
zone, the equation for hydrogen ion transport becomes (it is to be noted that the diffusivity for the cathode catalyst layer is different from the diffusivity of the cathode diffusion layer) : > : > : > = ": >,867 −% − G,867 D^ F : > , ! 2 2
(13)
where the last term denotes the reaction in the cathode diffusion layer. 2.25 Initial and Boundary conditions The concentration of protons and glucose in the fuel cell system follows a system of linear partial differential equation. Hence, both initial and boundary conditions are required to obtain a solution. The glucose concentration in the cell domain is initially assumed to be 0 _, with glucose entering from the left of the anode compartment, with a constant feed concentration at the beginning of computation time. Therefore, At ! = 0,
89 :;< =9 = 0 and : > = 0, ∀ 2
At ! > 0
bc
89 :;< =9 = 89 :;< =9 and : > = 0, at 2 = 0
(14)
(15)
Also, the concentrations at interfaces of all layer are assumed to be in equilibrium. Therefore,
d 8
9 :;< =9
e
L/Lf gL
= d 89 :;< =9 eL/L gL
(16)
= d : > |L/Lf gL
(17)
where 2 is the coordinate of each interface. Similarly, At 2 = 4[8 ,
d
: > |L/Lf gL
f
i : > = 0, ∀ ! i2
(18)
2.26 Operating Temperature The behavior of fuel cell is strongly affected by the change in operating temperature, as the rate constants and diffusivities are functions of temperature. The rate constants are assumed to vary as per Arrhenius equations and diffusivity is assumed obey Eyring’s equation. ?l
" = "j k ℜ] 6
(19)
n = nj k
? P ℜ]
(20)
2.27 Current Density The current density is calculated at the cathode interface. The current density is a function of hydrogen ion gradient, and can be written as,
i : > p i2 L/LOPQqrls being the coordinate of the cathode plate. o = −( · ": > d
2BItuc
(21)
2.3 Solution methodology For obtaining concentrations of hydrogen ions and glucose, a system of partial differential equations (PDEs) has to be solved. The resulting system is linear, but cannot be solved analytically without complex mathematical treatment. Therefore, it was solved numerically, by reducing the system of PDEs to a system of ordinary differential equations by employing finite difference methods for reducing derivative terms by discretizing the domain into finite nodes. The concentrations were then defined for these discrete nodes. Each layer was divided into 20 nodes, two equations being solved at each node, one for concentration hydrogen ions and other for concentration of glucose molecule. Commercial solver ‘ode15s’ which comes bundled with MATLAB was employed to solve the system. The cathodic current density was calculated using second order finite difference to approximate the derivative of concentration of hydrogen ions. 3.0 Results and Discussion Figure 3 shows the variation of the concentration of glucose with time across the anode diffusion layer, the enzyme layer and the membrane as a function of inlet glucose concentration and inlet flow rate. It is to be noted that the first layer in the figure denotes the anode diffusion layer, followed by the enzyme layer, the membrane, the cathode catalyst layer and the cathode diffusion layer. At initial values of time, the diffusional limitations in the anode diffusion layer causes a steep drop in the concentration profile of the glucose across the ADL. However, with increasing time, the concentration profile across the ADL increases and levels off. The reaction in the enzyme layer is indicated by the steep drop in concentration of the glucose across the enzyme layer. The parameter values listed in Table 1 were used for simulations presented in Figure 3 to 12.
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The transient variation of hydrogen ions can be seen in Figure 4 across the anode diffusion layer, the enzyme layer, the membrane, the cathode diffusion layer and the cathode catalyst layer. Complete consumption of hydrogen ion is assumed in the cathode catalyst layer and cathode diffusion layer. It can be seen that the concentration of the hydrogen ions is seen to increase in the enzyme layer where the glucose oxidation reaction happens but is seen to decrease after that due to consumption of the hydrogen ions in the cathode catalyst layer. Due to the presence of potential across the cell, the hydrogen ions experience a backward diffusive flux, which is reflected in Figure 4. Figure 5 shows the variation of glucose concentration across the anode diffusion layer, the enzyme layer and the membrane as a function of inlet glucose flow rate. There is no drop in the concentration of glucose species across the anode diffusion layer since convective flux due to high flow rates overcome the low diffusive fluxes.
There is a steep drop in the glucose
concentration in the enzyme layer, due to reaction with glucose oxidase. Figure 6 shows the variation of glucose concentration across the ADL, EL and the membrane as a function of temperature. Since the diffusion coefficients increase with increase in temperature, there is a notable increase in diffusive flux with temperature in the ADL. The concentration of the glucose is seen to drop steeply across the enzyme layer indicating that the reaction rate is a strong function of temperature. Operating temperature also plays a major role in determining consumption rate of glucose molecules, evident by steeper drop in the concentration of glucose in enzyme layer. Figure 7 shows the variation of hydrogen ions across the ADL, EL, membrane and the CCL as a function of temperature. The trend in hydrogen ion concentration can be deduced from Figure 6, as higher operating temperature results in higher concentration of hydrogen ions in the enzyme layer. Figure 8 shows the concentration of hydrogen ions across the enzyme layer as a function of the thickness of the enzyme layer. It is seen that the higher the thickness of the enzyme layer, the higher the concentration of the hydrogen ions. More thickness of enzyme layer provides larger region and more sites for oxidation, resulting in increased generation of hydrogen ions. Figure 9 shows the polarization curve of the enzymatic glucose fuel cell as a function of time. The system achieves steady state in about 2 minutes, for given set of parameters. Figures 10 and 11 shows 8
the polarization curve of the enzymatic glucose fuel cell as a function of flow rate and temperature at steady state. The inlet concentration of glucose molecules strongly affects the polarization curves and cell performance. Figure 12 shows the polarization curve of the enzymatic glucose fuel cell as a function of the thickness of the enzyme layer. The performance of the fuel cell is seen to increase with increasing thickness of the enzyme layer. Figure 13 shows the comparison of modeling results with experimental data13. Simulation results are compared with experimental polarization curves obtained at 310 K and flow rate of 167 mm3/s. The results obtained from the model were found to compare well with the experimental data. Table 1 enlists the parameters used in carrying out simulations presented in Figure 13, with aforementioned temperature and flow rate values.
4. Conclusion A mathematical model is developed to study the concentration profiles of glucose and hydrogen ions across the enzymatic glucose fuel cell. The concentration profiles of glucose are studied across the anode diffusion layer, enzyme layer and the membrane. The concentration profiles of glucose are studies across the anode diffusion layer, enzyme layer, membrane and the cathode catalyst layer. The concentration profiles are found to be sensitive to temperature and inlet flow rate. The thickness of the enzyme layer is found to play a significant role in determining the concentration of hydrogen ions across the layer and in the overall performance of the cell. Model results are compared with experimental data and found to agree very well.
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5.Figure Captions Figure 1 - Schematic diagram of a enzymatic glucose fuel cell representing various layers and their thicknesses. Figure 2 – Schematic diagram showing the modeling configuration of a enzymatic glucose fuel cell. Figure 3 - Variation of glucose concentration as a function of time in the fuel cell domain Figure 4 - Variation of concentration of hydrogen ions as a function of time in the fuel cell domain Figure 5 - Glucose concentration across the cell with varying flow rates. Figure 6 - Effect of temperature on glucose concentration profile across the cell. Figure 7 - Hydrogen ion concentration across the cell with varying operating temperatures. Figure 8 - Hydrogen ion concentration inside the enzyme layer as a function of the thickness of the enzyme layer. Figure 9 –Transient polarization curves Figure 10 – Polorization curves of the fuel cell with varying glucose feed flow rates. Figure 11 – Polorization curves of the fuel cell with various operating temperatures. Figure 12 – Polorization curves of the fuel cell as a function of thickness of enzyme layer Figure 13 - Comparison of polarization curves with experimental data.
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6.Acknowledgements The authors wish to acknowledge BITS Pilani for enabling us to submit this article. 7. References 1.Vladmir R, Lea M, Physical modeling of enzymatic glucose fuel cells,
Advances in Chemical
Engineering and Science, 2013, 3, 218-224 2.Vladmir R, Current Voltage modeling of enzymatic glucose fuel cells, Advances in Chemical Engineering and Science, 2015, 5,164-170. 3.Gymama S , Tanmay Kulkarni, Enzymatic glucose fuel cell and its application, Biochip Tissue Chip, 2015, 5, 1-6. 4.Atanassov P, Chris A and Shelley M, Enzymatic bio fuel cells, Electrochemical Society Interface, 2007, 28-32. 5.Peter C, Donal Leech, Fully enzymatic glucose oxygen fuel cell, Anal. Chem, 2016,88, 2156-2163. 6.Ranoo P and Basu.S, Mathematical modeling and experimental verification of direct glucose anion exchange membrane fuel cell, Electrochimica Acta, (2013) 113 : 42– 53.
7.Sheng C, Der.J , Shing W, Dynamic Modeling of Anode Function in Enzyme-Based Biofuel Cells Using High Mediator Concentration Energies 2012, 5, 2524-2544;
8.Yin S, Varun P, Chunlei W, Modeling and Simulation of Enzymatic Biofuel Cells with Three-Dimensional Microelectrodes, Energies, 2014, 7, 4694-4709. 9.Basu D, Basu S, Mathematical modeling of overpotentials of direct glucose alkaline fuel cell and experimental validation, J. Solid State Electrochem, 2013, 17, 2927-2938. 10.J.J.Aneppu, Masters thesis, Mathematical simulation of direct glucose fuel cells, Governors State University, Fall 2015. 11. John Newman and K.E.Thomas Alyea, Electrochemical Systems, Wiley, 2004. 12. S.Phul, A.Despande and Balaji Krishnamurthy, The effect of potential drop on the capacity fading in lithium ion batteries, Electrochimica Acta, 164 (2015), 281-287.
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8. Nomenclature
v ( w"4 4 _kx 4 "4 ℜ bc ! 2 " ,A % & ( ' ∇Φ + ,z o 2{ GBI G D^ F DF c B 4A 4H?5 "j nj
absolute temperature fuel cell anode diffusion layer enzyme layer/anode catalyst layer membrane cathode catalyst layer cathode diffusion layer universal gas constant concentration of component ‘y’ inlet feed concentration of component ‘y’ time co-ordinate on axis along length of the fuel cell diffusion coefficient of component ‘y’ in medium ‘x’. velocity or volume flux vector charge on species of component ‘y’ Faraday’s constant mobility of species ‘y’ electric field rate of formation of component ‘y’ in reaction ‘o’ total number of components current density co-ordinate of interface between two layers enzyme rate constant rate constant for cathode reaction concentration of oxygen gas concentration of glucose oxidase enzyme activation energy for diffusion per mole activation energy for reaction per mole length of layer ‘x’ length of membrane electrode assembly maximal diffusion constant maximal reaction rate constant
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