Convection–diffusion–reaction inside a porous sphere under oscillatory flow including external mass transfer

Prakash, Jai; De, Sirshendu; Sekhar, G. P. Raja

doi:10.1088/0169-5983/43/1/015508

Cited by 6 publications

(4 citation statements)

References 33 publications

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Section: Stirred Tank Systemmentioning

confidence: 99%

“…Under these conditions, the average concentration in the pellets depends only on the radial position. The reader interested in the coupling of convection, diffusion, and reaction inside permeable cylindrical and spherical porous pellets in an oscillatory flow using Robin-type boundary condition is referred to the works of Prakash et al 21,22 It should be stressed that many times the structure of the governing equation for the fluid in the tank is guessed on the basis of intuition and experience. However, this equation can be obtained from eqs 1b−3b by a much more rational upscaling procedure.…”

Section: ■ Stirred Tank Systemmentioning

confidence: 99%

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Effect of Reaction and Adsorption at the Surface of Porous Pellets on the Concentration of Slurries

Sales‐Cruz

Valdés‐Parada

Goyeau

et al. 2012

Ind. Eng. Chem. Res.

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In this work, we analyze the influence of interfacial reaction and adsorption processes over the predictions of the unsteady average macroscopic concentrations in slurries. The isothermal mass transport model consists of the equations for the well-mixed fluid and the one corresponding to the pellets. The latter consist of porous media where first-order reaction kinetics and linear adsorption take place. The interfacial flux boundary condition considers the mass transfer resistance in the fluid, as well as superficial reaction and accumulation terms. The model solution is carried out by means of Fourier series expansions. It is shown that, unless the adsorption and reaction at the surface of the porous pellets are much more intense than in the bulk, the error introduced by not considering these interfacial phenomena is negligible. The results from this work are relevant for improving design and control applications of slurries involving fast reactions or strong adsorption.

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Section: Stirred Tank Systemmentioning

confidence: 99%

Section: ■ Stirred Tank Systemmentioning

confidence: 99%

Effect of Reaction and Adsorption at the Surface of Porous Pellets on the Concentration of Slurries

Sales‐Cruz

Valdés‐Parada

Goyeau

et al. 2012

Ind. Eng. Chem. Res.

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“…Theoretically, the mass transport processes in a porous slab and a porous sphere, coupled with convection and reaction kinetics, were thoroughly discussed. − For the first order kinetics, the effectiveness factor (η), introduced to take account of the influence of pore diffusion on reaction rate, is given as a function of Thiele modulus for spherical particle in the case of constant concentration at the surface − where ϕ s is the Thiele modulus for a sphere, k 1 is first-order kinetic constant (s –1 ), R is the sphere radius (cm), and D e is the effective diffusion coefficient (cm 2 s –1 ). Based on eqs and , the more effective diffusion coefficient ( D e ) and shorter sphere radius ( R ) are favorable for achieving better effectiveness factor (η).…”

Section: Introductionmentioning

confidence: 99%

Hollow Mesoporous Organic Polymeric Nanobowls and Nanospheres: Shell Thickness and Mesopore-Dependent Catalytic Performance in Sulfonation, Immobilization of Organocatalyst, and Enantioselective Organocascade

Xie

Wei

Zhang

et al. 2019

Ind. Eng. Chem. Res.

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Heterogeneous asymmetric multicomponent/multicatalytic organocascade faces the enormous challenges of tedious immobilization of catalysts, mass transfer, and stereoselective control. In this paper, the mesopore-abundant and well-shaped hollow mesoporous organic polymers, nanobowls (HMOPBs) and nanospheres (HMOPSs), are fabricated via emulsion polymerization of styrene on a polystyrene (PS) core and then removal of PS, accompanied by the adsorption of Co2+ ions, transformation of Co2+ into Co(OH)2, and final removal of Co(OH)2. Among them, the nanobowl HMOPBs(a) with hollow interiors, mesoporous shells, and thin shell thickness (16 nm) possesses the largest surface area (185.7 m2 g–1) and displays the highest reaction kinetics in the sulfonation (2.85 mmol H+ g–1) and then immobilization of 9-amino(9-deoxy)epi-quinine (QNNH2, 1.35 mmol g–1). For the 2,4-substituted bulky reactants in an asymmetric double-Michael cascade, the as-fabricated functional nanobowl can provide a suitable microenvironment to meet the requirements of the good to excellent double-Michael organocascades, which originates from its thin shell thickness and mesopore-dependent shell.

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“…Further, they [24,25] have also studied the problem of convection-diffusion-reaction in case of a circular cylindrical pellet/ spherical pellet subject to Dirichlet condition/ Robin condition. They have observed a significant effect of oscillation on the nutrient transport inside the porous pellet.…”

Section: Introductionmentioning

confidence: 99%

Convection–diffusion–reaction inside a permeable cylindrical porous pellet under oscillatory flow: the effect of Robin boundary condition

Prakash

Sekhar

2011

Int J Adv Eng Sci Appl Math

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A semi analytical solution is presented considering the interaction of mass transfer with a homogeneous chemical reaction of zero order/ first order reaction inside a cylindrical porous pellet. The corresponding hydrodynamic problem is formulated as a problem of flow past a porous circular cylinder for Stokes-Darcy coupled system. This is solved using a stream function approach employing the continuity of pressure, continuity of normal velocity component and Saffman slip condition for the tangential velocity component at the porous-liquid interface. The velocity field obtained inside the porous pellet is used to study the combined convection-diffusion-reaction problem subject to Robin type boundary condition, which takes into account the external mass transfer resistance. It is seen that in case of zero order reaction, for a particular combination of physical parameters, concentration takes negative values at some points inside the pellet, which is generally termed as starvation. A necessary and sufficient condition is derived ensuring the non-negativity of the concentration inside the pellet. List of symbolsa Radius of the porous pellet [m] k Permeability of the porous pellet [m 2 ] r Radial distance v e Oscillatory velocity external to the porous pellet [m/s] p e Oscillatory pressure external to the porous pellet [N/m 2 ] V e Amplitude of the oscillatory velocity external to the porous pellet [m/s] P e Amplitude of the oscillatory pressure external to the porous pellet [N/m 2 ] V i Velocity internal to the porous pellet [m/s] P i Pressure internal to the porous pellet [N/m 2 ] p 0 Constant [N/m 2 ] U 1 Magnitude of the far field uniform velocity [m/s] c i Concentration inside the porous pellet [mol/m 3 ] S Uptake rate [mol/s] k 0 Rate constant [s -1 ] D Diffusivity [m 2 /s] I n Modified Bessel function of first kind K n Modified Bessel function of second kind k m Mass transfer coefficient [m/s] c 0 Concentration at the surface of the porous pellet [mol/m 3 ] c Dimensionless concentration Da ¼ k a 2 Darcy number l ¼ ffiffiffiffiffiffi Da p Dimensionless parameter Pe ¼ U 1 a D Péclet number Sh ¼ k m a D Sherwood number Re ¼ lU 1 a q Reynolds number Sc ¼ m D

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Convection–diffusion–reaction inside a porous sphere under oscillatory flow including external mass transfer

Cited by 6 publications

References 33 publications

Effect of Reaction and Adsorption at the Surface of Porous Pellets on the Concentration of Slurries

Effect of Reaction and Adsorption at the Surface of Porous Pellets on the Concentration of Slurries

Hollow Mesoporous Organic Polymeric Nanobowls and Nanospheres: Shell Thickness and Mesopore-Dependent Catalytic Performance in Sulfonation, Immobilization of Organocatalyst, and Enantioselective Organocascade

Convection–diffusion–reaction inside a permeable cylindrical porous pellet under oscillatory flow: the effect of Robin boundary condition

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