New Analytical Expressions of Concentrations in Packed Bed Immobilized-Cell Electrochemical Photobioreactor
Abstract
:1. Introduction
2. Formulation of the Problems
3. Analytical Expression of the Concentrations Using Akbari-Ganji’s Method
3.1. Concentration of Glucose (Substrate)
3.2. Concentration of Hydrogen (Product)
3.3. Normalized Steady-State Source Terms of Liquid and Gas Phases
4. Analytical Expression of the Concentrations Using Taylor Series Method
4.1. Concentration of Glucose (Substrate)
4.2. Concentration of Hydrogen (Product)
4.3. Normalized Steady-State Source Terms of Liquid and Gas Phases
5. Comparison of Analytical Results with Previous Analytical Results and Numerical Simulation
6. Discussion
7. Conclusions
Supplementary Materials
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
List of Symbols
Symbols | Description | Units |
Local substrate concentration | kg/m3 | |
Radius of the gel granule | m | |
Diffusion coefficient | m2/s | |
Absolute permeability | m2 | |
Molecular weight | kg mol−1 | |
Photobioreactor height | m | |
Maintenance coefficient | h−1 | |
Source item in mass conservation equation | ||
Source item in species conservation equation | ||
Specific area | m2/kg | |
Growth associated kinetic constant for hydrogen production | None | |
Non- growthassociated kinetic constant | h−1 | |
Specific growth rate | h−1 | |
Maximum specific growth rate | h−1 | |
Dimensionless substrate concentration | None | |
Dimensionless product concentration | None | |
Dimensionless parameter | None | |
Dimensionless parameter | None | |
Dimensionless parameter | None | |
Dimensionless parameter | None | |
Dimensionless parameter | None | |
Dimensionless distance | None | |
Superscripts | ||
S | Substrate | |
H2 | Hydrogen | |
CO2 | Carbon dioxide | |
Subscripts | ||
G | Gas phase | |
l | Liquid phase | |
Abbreviation | ||
AGM | Akbari-Ganji’s method | |
ADM | Adomian decomposition method | |
TSM | Taylor series method | |
HPM | homotopy perturbation method | |
PM | perturbation method |
Appendix A. MATLAB Code for Numerical Solution of the Nonlinear Equations (9) and (10)
- function pdex4
- m = 2;
- x = linspace(0,1);
- t = linspace(0,10);
- sol = pdepe(m,@pdex4pde,@pdex4ic,@pdex4bc,x,t);
- u1 = sol(:,:,1);
- u2 = sol(:,:,2);
- %------------------------------------------------------------------
- figure
- plot(x,u1(end,:))
- title(‘u1(x,t)’)
- xlabel(‘Distance x’)
- ylabel(‘u1(x,1)’)
- %------------------------------------------------------------------
- figure
- plot(x,u2(end,:))
- title(‘u2(x,t)’)
- xlabel(‘Distance x’)
- ylabel(‘u2(x,2)’)
- % -----------------------------------------------------------------
- function [c,f,s] = pdex4pde(x,t,u,DuDx)
- c = [1; 1];
- f = [1; 1].* DuDx;
- a1 = 10; p1 = 30; p2 = 0.1; r1 = 0.1; r2 = 0.1; r3 = 0.1; r4 = 0.1; %
- F1 = −(((p1+r1)*u(1) + r2)/(1 + a1*u(1)));
- F2 = −(((p2+r3)*u(1) + r4)/(1 + a1*u(1)));
- S = [F1; F2];
- % -----------------------------------------------------------------
- function u0 = pdex4ic(x)
- u0 = [0; 0];
- % -----------------------------------------------------------------
- function [pl,ql,pr,qr] = pdex4bc(xl,ul,xr,ur,t)
- pl = [ul(1)-0; ul(2)-0];
- ql = [1; 1];
- pr = [ur(1)- 1; ur(2)-1];
- qr = [0; 0];
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Jeyabarathi, P.; Abukhaled, M.; Kannan, M.; Rajendran, L.; Lyons, M.E.G. New Analytical Expressions of Concentrations in Packed Bed Immobilized-Cell Electrochemical Photobioreactor. Electrochem 2023, 4, 447-459. https://doi.org/10.3390/electrochem4040029
Jeyabarathi P, Abukhaled M, Kannan M, Rajendran L, Lyons MEG. New Analytical Expressions of Concentrations in Packed Bed Immobilized-Cell Electrochemical Photobioreactor. Electrochem. 2023; 4(4):447-459. https://doi.org/10.3390/electrochem4040029
Chicago/Turabian StyleJeyabarathi, Ponraj, Marwan Abukhaled, Murugesan Kannan, Lakshmanan Rajendran, and Michael E. G. Lyons. 2023. "New Analytical Expressions of Concentrations in Packed Bed Immobilized-Cell Electrochemical Photobioreactor" Electrochem 4, no. 4: 447-459. https://doi.org/10.3390/electrochem4040029