ORIGINAL RESEARCH Convection–diffusion–reaction inside a permeable cylindrical porous pellet under oscillatory flow: the effect of Robin boundary condition Jai Prakash • G. P. Raja Sekhar • Sirshendu De Published online: 16 August 2011 Ó Indian Institute of Technology Madras 2011 Abstract A semi analytical solution is presented con- sidering the interaction of mass transfer with a homoge- neous 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 inter- face. 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. Keywords Stokes flow  Darcy’s law  Saffman condition  Oscillatory flow  Nutrient transport  Starvation zone  External mass transfer List of symbols a 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 Pe ´clet number Sh ¼ k m a D Sherwood number Re ¼ lU 1 a q Reynolds number Sc ¼ m D Schmidt number J. Prakash  G. P. Raja Sekhar (&) Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur 721302, India e-mail: rajas@maths.iitkgp.ernet.in J. Prakash e-mail: jp@maths.iitkgp.ernet.in S. De Department of Chemical Engineering, Indian Institute of Technology Kharagpur, Kharagpur 721302, India e-mail: sde@che.iitkgp.ernet.in 123 Int J Adv Eng Sci Appl Math (March–December 2011) 3(1–4):60–70 DOI 10.1007/s12572-011-0030-2 IIT, Madras