Effect of Biot number on unsteady reaction-diffusion phenomena and analytical solutions of coupled governing equations i
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pISSN: 0256-1115 eISSN: 1975-7220
INVITED REVIEW PAPER
INVITED REVIEW PAPER
Effect of Biot number on unsteady reaction-diffusion phenomena and analytical solutions of coupled governing equations in porous particles with various shapes Young-Sang Cho† and Sohyeon Sung Department of Chemical Engineering and Biotechnology, Korea Polytechnic University, 237 Sangdaehak-ro, Siheung-si, Gyeonggi-do 15073, Korea (Received 20 February 2020 • Revised 3 June 2020 • Accepted 7 July 2020) AbstractAnalytical solutions of transient concentration profile inside particles were calculated by solving rection-diffusion equation in spherical, cylindrical, and slab-type porous particles, assuming first order irreversible reaction. Timedependent concentration in bulk fluid phase was assumed as exponential decay for each particle, and convective boundary conditions were taken using arbitrary Biot number to obtain a general solution by eigenfunction expansion or Laplace transform method. Factors affecting average transient concentration inside particles were studied by adjusting Biot number, reaction time, and Thiele modulus as well as the position inside particles. To predict transient bulk concentration in batch photocatalytic reactor containing porous particles, coupled differential equations were solved by Laplace transform to obtain analytical solutions of bulk concentration as well as average concentration inside porous particles as a function of reaction time. The factors affecting transient concentrations were investigated by adjusting the concentration and porosity of the catalytic particles, morphology of the particles, and Thiele modulus in batch-mode reactor to study reduction speed of reactant concentration during photocatalytic reaction. The solutions from coupled differential equations were useful for the prediction of transient behavior in batch-type photocatalytic reactor and were compared with the results from CSTR containing slab-type photocatalytic particles with various space time. Keywords: Porous Particles, Reaction-diffusion Equation, Eigenfunction Expansion, Biot Number, Time-dependent Boundary Condition, Coupled Differential Equations, Laplace Transform
Although numerical simulation is a powerful means for these research topics, mathematical analyses are still challenging for treatment of reaction-diffusion equation subject to arbitrary boundary conditions on a particle surface to get approximate or exact solutions by inverse Laplace transform including our previous results [10-12]. Unlike conventional reaction-diffusion equation, generation of chemical species by first-order reaction has been also studied by eigenfunction expansion method, which is useful degradation behavior of PLGA particles [13]. Unlike mathematical analyses on conventional porous particles, porous spherical particles with inert cores have been also treated for reaction-diffusion phenomena in batch or packed bed reactor by Laplace transform [14]. However, further researches are still necessary, since previous studies on analytical or appr
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