In this paper the results of previous researches are extended and equations developed by means of which the resistance to fluid flow offered by a bed of spherical or non-spherical material may be computed with reasonable accuracy. The equations cover a range of porosity of the bed from 30 per cent to 90 per cent and practically the whole of the technically important range of Reynolds number, while particle shapes ranging from spheres to hollow cylinders, shell insulators and wire nails are covered. A new curve for wall-effect, depending upon Reynolds number, is also included. It is believed that the present work is of more general application than any which has previously been suggested.
The resistance coefficient–Reynolds number relationship for a bed of spherical particles, closely graded about a mean size, is considered and it is shown that this relationship correlates with that for a bed of non-spherical particles, provided a suitable allowance for particle shape is made. It is shown that for very large or very small values of the Reynolds number the co-ordinates of the resistance coefficient–Reynolds number curve fall closely about a mean line but for the transition-range the points suffer considerable scatter—probably due to the indeterminate nature of the flow in the pore channels for this range of the Reynolds number. Curves are given showing the frequency of occurrence of variations in the resistance coefficient, and an example of the practical application of these curves to a design problem is fully worked out. Curves relating the resistance coefficient to the Reynolds number for beds of particles of various degrees of angularity are put forward tentatively, these relationships being, as yet, not completely established.
A general equation governing the flow of liquids through a bed of granular material is derived by the use of the method of dimensional analysis. The equation so obtained has been verified by experiment for liquids having absolute viscosities ranging from approximately 16·0 poises to about 0·01 poise, the bed material being spherical shot closely graded about a mean size. Examination of the experimental results shows that the curve relating fluid resistance to Reynolds number has a form very similar to that for a single sphere moving in an infinite fluid, although the discontinuities which are present in the latter curve at the critical value of the Reynolds number are absent in the former; that the effects of the walls of the container are negligible; that the resistance to flow is accurately proportional to depth of bed, and that the hydraulic resistance varies inversely as the fourth power of the voidage of the bed approximately, for beds of normal density of packing, although the value of the index varies with the value of the voidage. The evidence also indicates that the hydraulic resistance is not accurately defined by the mean value of the voidage, but is subject to considerable variations due to the statistical nature of a bed, and that the effect of the walls of the container may be computed on the assumption of flow through a pair of parallel channels of different hydraulic resistance.
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