65 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 ABPP PD = - is required and our goal is to relate PD 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.66 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, volume of particles6External surface area of particlespD = × (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 L Face A Face B q67 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. ( )24VQ Dp=. (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:68 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). ( )AB ABsin P PLgrq- = - + -PP (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