A criterion to avoid starvation zones for convection–diffusion–reaction problem inside a porous biological pellet under oscillatory flow

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

doi:10.1016/j.ijengsci.2010.02.004

Cited by 11 publications

(8 citation statements)

References 28 publications

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“…1; i.e., neglecting external mass transfer resistance, Eq. 27 reduces to Dirichlet condition, i.e., c = 0 [23,24,29]. Note that we have omitted the symbol~from Eqs.…”

Section: Zero Order Reaction Casementioning

confidence: 99%

“…2, 3). Keeping similar physical configuration, however, considering Dirichlet boundary condition, Prakash et al [23,24] have studied the problem of convection-diffusion-reaction inside porous pellet of spherical/cylindrical geometries, under oscillatory flow. It is expected that due to oscillatory nature of the flow, the nutrient gets flushed out along the flow direction.…”

Section: Zero Ordermentioning

confidence: 99%

“…Prakash et al [23] extended the study of Stephanopoulos and Tsiveriotis by considering oscillatory Stokes flow. 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.…”

Section: Introductionmentioning

confidence: 99%

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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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“…1; i.e., neglecting external mass transfer resistance, Eq. 27 reduces to Dirichlet condition, i.e., c = 0 [23,24,29]. Note that we have omitted the symbol~from Eqs.…”

Section: Zero Order Reaction Casementioning

confidence: 99%

Section: Zero Ordermentioning

confidence: 99%

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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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“…It is well known that the mass transfer of a species is enhanced by several orders of magnitude when it is present in a fluid medium subjected to oscillatory motion (Prakash et al 2010b). There are a number of applications where oscillatory flow is used, such as waste water treatment, heat exchange, etc.…”

Section: Introductionmentioning

confidence: 99%

“…In the case of convection-diffusion-reaction inside porous spheres/cylinders, often it is seen that, corresponding to the zeroth-order reaction, the concentration takes negative values within particular zones. These are termed in the literature as 'starvation zones' or dead zones (see Stephanopoulos and Tsiveriotis 1989, Valdés-Parada et al 2005, Prakash et al 2010b. In order to have physically meaningful results, one has to classify the nutrient-rich and nutrient starvation zones.…”

Section: Introductionmentioning

confidence: 99%

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

Prakash

Sekhar

2011

Fluid Dyn. Res.

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This study aims to show the effect of oscillation and external mass transfer on nutrient transport inside a porous sphere when the flow external to the porous sphere is of oscillatory nature. Unsteady Stokes equations are used for the flow outside the porous sphere and Darcy's law is used inside the sphere. We employ a complete general solution of oscillatory Stokes equations in order to solve the corresponding hydrodynamic problem. Then the convection-diffusion-reaction problem is formulated and solved analytically for both zeroth-and first-order rates of nutrient uptake. The Robin-type boundary condition is used to quantify the external mass transfer at the porous-liquid interface. Further, in the case of zeroth-order reaction, a general condition is derived between the Peclet number and the Thiele modulus to preclude the nutrient reduction everywhere inside the sphere. Nomenclature aradius of the porous sphere (m) k permeability of the porous sphere (m 2 ) r radial distance X position vector v e oscillatory velocity external to the porous sphere (m s −1 ) p e oscillatory pressure external to the porous sphere (N m −2 ) V e amplitude of the oscillatory velocity external to the porous sphere (m s −1

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Physics of unsteady Couette flow in an anisotropic porous medium

Karmakar

2021

J Eng Math

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A criterion to avoid starvation zones for convection–diffusion–reaction problem inside a porous biological pellet under oscillatory flow

Cited by 11 publications

References 28 publications

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

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

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

Physics of unsteady Couette flow in an anisotropic porous medium

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