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The 1-D axial, laminar, Newtonian flow through a tube of uniform, but noncircular, cross-section is analyzed. The tube boundary is r = R[1 + ef(θ)], in which R is a reference radius, e is a small parameter, and the function f(θ) of the polar angle is general, albeit subject to minimal constraints ensuring single-valuedness and quadrant symmetry. All results are determined as asymptotic expansions complete to terms O(e2), which include the velocity distribution, average velocity, volume flow rate, ratio of flow rates in the noncircular to that in a circular pipe of the same cross-sectional area, the friction factor-Reynolds number dependence, the permeability of packed beds comprising noncircular capillaries, kinetic-energy and momentum-flux correction factors, the capillary penetration rate under the influence of the capillary pressure and the equilibrium value of the capillary rise. Expressions are derived in terms of the general boundary function f(θ) and, for special cases, when f(θ) = sin2kθ (k = 1, 2, 3) and f(θ) = sin2kθ with k a positive integer. The results provide quantitative measures of the effect of tube shape on flow properties and on imbibition and drainage from noncircular capillaries.

This paper discusses settling rates, yield stresses, and shear-stress/shear-rate dependences which were determined as functions of solids concentration for suspensions in water of coal particles of narrow size fractions. Particles and suspensions are characterized by coal analyses, determination of heating value, solid heat capacity and thermal conductivity, particle size analyses, and determination of densities and maximum packing concentrations for each particle size range. Analysis of the settling data establishes that the usually strong retardation of (hindered) settling due to concentration is compounded by particle shape effects. Shear-stress/shear-rate data for concentrated suspensions of the three larger and relatively narrower sized particles over the broad range of shear rates from about 1 to 10{sup 4} s {sup {minus}1} suggest the behavior to be essentially Newtonian, albeit with a marked particle concentration augmentation. Suspensions of the smallest coal particles of broader size distribution were strongly non-Newtonian, possessing a yield stress at low shear, essentially power-law shear-thinning behavior over the intermediate range and approaching high-shear Newtonian limiting behavior above about 10{sup 3} s{sup {minus}1}.

In slurry transport, the critical velocity is defined as the minimum velocity demarcating flows in which the solids form a bed at the bottom of the pipe (bed load flows) from fully suspended flow. An analysis based on balancing the energy required to suspend the particles with that derived from dissipation of an appropriate fraction of the turbulent eddies is used to develop a correlation for prediction of the critical velocity. Comparison of the results with available experimental critical velocity data, relating to a rather wide variety of slurry systems, confirms that the present correlation does a superior job of prediction than all previously proposed critical velocity correlations.

A multiphase model for the flow of dense, noncolloidal, settling slurries through horizontal pipelines, and an associated Galerkin-finite element numerical scheme to carry out computer simulations based on the model, were developed. The principal elements of the model are: equations for estimation of the velocity distributions for each component; and those for the concentration distributions. In the former, which are derived from the local volume-averaged, time-smoothed conservation of mass and momentum equations, the total stress is comprised of components to account for interparticle and particle-boundary as well as viscous and turbulent interactions. The governing equations for the concentration distributions are derived similarly from the convective diffusion equation, with the eddy diffusivity modified to account for the presence of the solid particles. Numerical simulations and detailed comparisons with experiment for the flow of gypsum, coal, crushed glass, sand and gravel, covering particle sizes from 38.3 μm to 13,000 μm, and pipe diameters from 4 cm to 49.5 cm, were carried out. The agreement between model prediction and experiment is excellent.

Although multicomponent, liquid phase adsorption is often encountered in industrial application, most of the studies concerning fixed-bed sorption processes have been dealt primarily with systems containing one adsorbable species. The present work provides a numerical solution to the fixed-bed, multicomponent adsorption problem, taking into account both separately or in combination the effect of liquid and/or solid phase mass transfer resistance and of variation of the assumed governing adsorption isotherm. Numerical solutions of several example problems are presented and compared with results based on equilibrium theory and a simplified method developed by Cooney and co-workers previously (1972).

Pressure drop correlations for flow of slurries in pipelines were developed for each of the following four flow regimes: flow with a stationary bed, saltation flow, heterogeneous flow, and homogeneous flow. A total number of 2 848 data points, comprised of 1 912 collected from the published literature together with 936 taken using our own test pipelines and relating to ranges of the pertinent variables extensive enough to span all four flow regimes were used as the basis of these correlations. Also, these data were used in developing an associated quantitative regime delineation scheme. The correlations provide an improved predictive capability over previously available procedures and are also broader in scope. The delineation procedure developed here permits straightforward classification of the data according to the flow regime prevailing, and it is moreover inclusive of all the data and is self-consistent.

A simulation procedure for predicting the dynamic behavior of a deep bed filter over the entire practicable range of filter operation is developed. The method is based on synthesizing available quantitative results relating to filtration, and to porous media flows, within an overall framework which views the process to consist of two principal stages dominated by appropriate limiting deposition modes. Evaluation of the results through comparison with available data indicates, to the extent the nature of these types of data permits, that the method is surprisingly effective and even capable of predicting on satisfactorily quantitative basis some intricate details of observed filter behavior.

The use of tracer dispersion measurements in conjunction with associated pressure drop data, as an indirect diagnostic technique for the determination of particle deposit morphology in deep-bed filters, was investigated. The dispersion measurements consisted of the injection of an electrolyte tracer pulse at the inlet and the monitoring of the tracer peak as it traveled down the bed, while the pressure drop data consisted of the axial pressure gradient histories as deposition took place. These data are interpreted using dispersion and pressure drop theories established on the basis of assumed models of deposit morphology. The validity of this technique was confirmed experimentally.

A model is developed for predicting the sedimentation velocity in suspensions of multisized nonflocculating solids, in which the retarding effect of the smaller particles on the setting velocities of the larger ones is taken into account. Tests of the model, and comparisons with other models, demonstrate that it provides improved prediction of data on suspensions comprising both discrete particle size mixtures and continuous size distributions, and that it is applicable to continuous countercurrent solid-liquid operations.

A procedure for the application of the so-called h-transformation in solving fixed-bed multicomponent adsorption problems is developed. In order to apply this procedure, it is necessary to select an appropriate value for the transition-motion factor ξ, and the criteria for making this selection are established in this work.

The hydrodynamic drag force in a uniform flow field acting on a spherical body with small particles attached to its surface was determined experimentally. The experimental data are used to substantiate the approximate force expressions derived from available theories for several types of collector-particle geometries. Procedures for applying these results to the prediction of the pressure drop across a filter bed with various degrees of particle deposition are also developed.

The terminal velocities of spheres falling in aqueous solutions of hydroxyethyl cellulose and polyethylene oxide were determined with ruby and steel spheres. For each sphere the terminal velocity was obtained in seven cylinder sizes covering a range of sphere-to-cylinder diameter ratio from 0.0067 to 0.18. The data were used in an extrapolation to correct for the cylindrical wall effect. Several alternative extrapolation procedures for estimating the zero shear viscosity were attempted and the results were compared. An empirical correlation for the drag coefficient was developed in terms of the Ellis parameters of the fluids, and the Faxen wall correction for Newtonian flow. the correlation gives the drag coefficient within 4% for all cases in which the quantity ηvt/Dτ1/2 is less than 0.3.

A numerical method for the solution of the problem of steady state, incompressible Newtonian flow through periodically constricted tubes is developed. All terms of the Navier-Stokes equation are retained, including the nonlinear inertia terms. Sample calculations for a uniform periodically constricted tube, the geometry of which is connected with the modeling of a packed bed of sand are given, including streamlines, axial and radial velocity profiles, pressure profiles, and the dimensionless pressure drop versus Reynolds number relation. The effect of some geometric characteristics of periodically constricted tubes on their friction factor is investigated numerically, and comparison of some existing experimental data with calculated ones is made.

A new model for porous media comprised of monosized, or nearly monosized grains, is developed. In applying this model to a packed bed, the bed is assumed to consist of a series of statistically identical unit bed elements each of which in turn consists of a number of unit cells connected in parallel. Each unit cell resembles a piece of constricted tube with dimensions which are random variables. The problem of flow through each unit cell is reduced, subject to reasonable assumptions, to the determination of the flow in an infinitely long periodically constricted tube. The solution of this flow problem is given in a companion publication. This model, together with the solution of the flow through it, can be used for the modeling of processes which take place in the void space of a bed. As a preliminary test, theoretical friction factor values, based on the proposed model, were compared with experimental ones for two different beds and found to be in good agreement even in the region of high Reynolds numbers where the nonlinear inertia terms are significant.