Chemwol Engmeenn.e Srwnce Vol 35 &WI 1347-1356 Pergamon Press Ltd 1980 Pnnted ,n Great Bntam FLOW NON-UNIFORMITIES IN PACKED BEDS EFFECT ON DISPERSION R G CARBONELL Department of Chermcal EnsneerIng, Wnlverslty of Cahfornla, Davis, CA 95616, U S A (Recewed 29 January 1979) Abstract--A general equation IS denved for the dispersion coeflictent in a packed bed under condltlons of non-umform, one dlmenslonal flow Predlctlons of the theory are analyzed for the simple case of two conccntrx annular regions m a cylmdncal packed bed with dlffercnt void fractions Small differences m void fraction can lead to slgmficant changes m the dlsperslon coeffinent for the packed bed, both m lammar and turbulent flow reames The radial variations of the average residence time and spread of a pulse of non- reactive tracer are investigated 1 INTRODUCTION The existence of radial void fractmn dlstnbutlons m packed beds of particles IS well estabhshed Benenate and Brosllow El ] report void fractions m packed beds of spheres approachmg 1 near the wall and exhibiting a damped osctllatlon up to 9 particle diameters mto the bulk of the bed where the void fraction IS 0 4 Slmllar dlstrlbutlons have been measured by several other mvestlgators [24] Three very recent articles have focused attention on the importance of this wall effect on transport parameters m packed beds Schlunder [ 5 ] and Martin [6] were able to account for the extremely low values of the Nusselt numbers measured experrmentally m packed beds at particle Peciet numbers less than 10, by consldermg the void fraction m the packed bed as conslstmg of two separate reaons In a remon at the center of the bed whose area IS no less than 90% of the total bed area, the void fraction was considered constant at 0 40, while m the remamder of the bed, up to the wall, the void fraction was considered to have an average value of 0 50 The difference m fluld velocltles m the two regmes was enough to account for orders of magnitude differences m the Nusselt number at low Peclet numbers due to the importance of channeling at the wall at these low flow rates Botterlll and Denlaye [7] used a function that would describe how the void fraction would vary as a function of distance away from a cylmdncal rod embedded m a packed bed of spheres m order to properly take mto account the local flow velocltv m their estimates of heat transfer coefficients from the rod to the spheres Usmg thus model they obtamed much better agreement with the available data than was possible with models that considered the flmd velocity around the rods to be the same as the fluid velocity correspondang to the average bed void fraction One ISencouraged by the success of these mvestlgators to explore the consequences of this void fraction dlstrlbutlon on dlsperslon m packed beds This would improve our understandmg of dispersion m general and possibly allow for better interpretations of dispersion coefficients in packed bed reactors Lerou and Froment [S] and &hertz and Blschoff [9] measured velocity profiles m packed beds that clearly indicate regons of high flow rate near the wall, thus provldmg addltlonal experimental evidence as to the existence of the void dlstrlbutlon and its importance m packed bed reactors Our approach here IS to consider the case of one- dlmenslonal flow m a packed bed subject to a void fraction dlstributlon as described above The non- uruformlty m the void fraction will result m a non- uniform velocity profile that can be estimated from the Ergun equation [lo] Followmg the approach plo- neered by Taylor [ll 1, we obtam a general equation for the dispersion coefficient valid for any Aow non- uniformity As a specific example we consider the void fraction dlstrlbutlon chosen by Schlunder [5] and Martin [6] m then analysis Computations usmg this model indicate that these flow non-umforrmtles can have a drastic effect on the dlsperslon coefFiclentfor the packed bed The response of the bed to pulses of mert tracer are investigated m order to see how the mean residence time and spread of the pulse would vary as a function of radial posltlon 2 THEORY We begm our analysis by consldermg the mass transport equation for a non-reactive solute m a porous medium m terms of the mtrmslc phase- averaged concentration m the fluid phase <c>‘, and the mtrmslc phase average velocity <v>=, a<c>= 7 + <v>= v<c>= = v (II) v <c>“) (1) The development of this equation has been considered in detail by Gray 1121. Slattery [13] and Whitaker [14,15], usmg volume-averagmg tech- niques The tensor B 1s a hspersion tensor that describes the local dispersion of the solute m the a phase anywhere m the bed Of course, the components of Q are functions of the local void fraction and local velocity m the b( phase In eqn (1) we have assumed the I347