# TAMU CHEN 304 - L7 (11 pages)

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- Pages:
- 11
- School:
- Texas A&M University
- Course:
- Chen 304 - Chem Engr Fluid Ops

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CBE341 Lecture 8 9 27 Dimensional analysis of flow in porous media and packed beds Processes designed to efficiently contact fluid with particles often consist of close packed particles constrained within a container Common examples include chemical reactors granular filters for waste water treatment and chromatography columns for analytical and preparative separations The figure on page 66 shows a cylindrical tube of diameter D packed with particles of equivalent diameter Dp D Dp equivalent diameter is defined below The particles are fixed in space and held in position through screens placed at either end of the tube The tube itself is in an inclined configuration such that its axis is at an angle q to the horizontal Consider an incompressible viscous fluid flow through this pipe entering at face A and exiting at face B at a volumetric flow rate of Q To sustain this flow a pressure drop DP PA PB is required and our goal is to relate DP to Q and the other relevant parameters In packed beds the external surface area of the particles is typically much larger than the internal surface area of the container therefore the resistance for the fluid motion comes predominantly from the particles and one can neglect the resistance offered by the tube walls Thus the tube diameter is not a relevant variable the typical rule of thumb is that the wall effects in a packed bed can be ignored when the tube diameter is greater than ten times the particle diameter D 10Dp In practical applications the particles are not perfectly spherical yet it is convenient to think in terms of roughly spherical particles How should we estimate the diameter of a spherical particle which is roughly equivalent to the actual non spherical particles employed in the packed bed Such questions play an important role in engineering as we are often forced to make rational simplifications and use information available in nearly similar systems to solve a given problem 65 q Face B L Face A Note that the flow of the fluid is resisted by the external surface of the particle which is in contact with the fluid The particles also block some volumes forcing the fluid to flow through the interstices Thus both particle volume and external surface area are important Consequently one demands that the idealized system with equivalent spherical particles occupy the same volume as the real particles and also have the same external surface area This argument leads to an equivalent diameter Dp 6 volume of particles External surface area of particles 92 One can readily verify that this definition recovers the correct particle diameter for the case of spherical particles Let us now define some variables that we encounter when we deal with beds of particles Bed voidage e is defined as the fractional bed volume available for the fluid flow In practical systems the particles may be porous however the pores inside the particles are generally not available for the pressure driven flow as they would be very resistive due to the large surface area of these pores Thus one is usually concerned only with the 66 void space available outside the particles We will assume in our analysis that the bed voidage is uniform everywhere in the bed Fluid velocity Here we have a couple of choices One can take the volumetric flow rate of the fluid Q and divide it by the total cross sectional area of the packed bed and get a velocity Such a velocity is commonly referred to as the superficial velocity which we will denote by V V Q p D 2 4 93 Alternately one can divide the volumetric flow rate by the cross sectional area available for fluid flow and obtain a velocity which is usually referred to as the average interstitial velocity v Clearly V v is simply equal to the fraction of the cross section available for flow which in a sufficiently large and homogeneously packed bed should be simply equal to the bed voidage e V v e 94 We will take V as the convenient velocity for the analysis and note that the entire analysis below can be repeated easily in terms of the interstitial velocity As in the case of flow through an empty pipe that we discussed earlier one can expect some entrance and exit effects in packed beds as well Away from the entrance and the exit one can expect the pressure to vary linearly along the pipe just as in the case of flow through empty pipes Typically the entrance and exit lengths are of the order of a few equivalent diameters of the packing particles as the particles are generally very small compared to the overall length of the bed one can neglect the end effects and assume to a good approximation that the pressure varies linearly with position everywhere in the bed As in the case of fully developed flow through an empty pipe we can construct mass and linear momentum conservation equations over any length L of the pipe where L can be the length of the entire bed as in the sketch or any partial length This will lead to the following conclusions 67 a The superficial velocity and the average interstitial velocity are independent of axial position in the bed b As the fluid flows through the packed bed the surface of the packing particles will exert a shear stress on the fluid resisting the flow This resistance captured by the average shear stress due to the packing per unit external area of the packing t can be related to the pressure drop and the gravitational force acting on the fluid through the application of linear momentum balance This analysis reveals that the relevant pressure drop that drives the flow is not the total pressure drop but the frictional pressure drop same as dynamic pressure drop PA PB Lr g sin q PA PB 95 Here r is the fluid density When the frictional pressure drop is zero the actual pressure variation is simply hydrostatic and there will be no flow To sustain flow the frictional pressure drop must be non zero We now consider how one can correlate the pressure drop per unit bed length DP L with various parameters involved in the problem To guide us in this analysis we again resort to the Buckingham Pi theorem DP L f r V Dp e 96 Here is its viscosity Applying the Buckingham Pi theorem we can conclude that Dp DP L f r VDp e r V2 97 This is analogous to flow through smooth empty pipe except that we now have an additional parameter e The existence of this extra parameter makes the correlation of 68 experimental data for the friction factor considerably more difficult This has prompted researchers to construct early on a simple

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