Search

The penetration rate of power-law non-Newtonian liquids in a circular tube driven by the capillary pressure is derived. The dimensionless fractional height of liquid in the tube relative to the equilibrium height, χ = h(t)/h(∞), is calculated as a function of an appropriately chosen dimensionless time, permitting expression of the height-time dependence as a function of the reciprocal of the exponent in the power-law model, s = 1/n. It is found that the rate of penetration and, therefore, also withdrawal of liquid is dramatically retarded as the liquid becomes more strongly shear-thinning, i.e., as n becomes small relative to unity.

The properties, settling rates, and the rheology of coal–water mixtures (CWM) made up from different coals were investigated. Test suspensions containing different concentrations of suspended particles were made up from each of four different pulverized parent coals: a Pittsburgh Seam No. 8 coal, an Illinois coal, and the product and feed from a flotation process using an Illinois coal. All the parent coals had broad particle-size distributions. In addition, the Pittsburgh Seam No. 8 coal was classified into four different size fractions using sieving, resulting in three coarser, narrow particle size cuts and the fine residual fraction passing 325-mesh, which had a broad size distribution. The property characterizations for all coals included the following: elemental analyses, heating values, particle size analyses, particle surface areas and pore sizes, solid heat capacities, and thermal conductivities. Tests on the suspensions included detailed supernatant ion analyses, and measurements of pH values and zeta potentials. For each test coal maximum-packing volume fractions were estimated using centrifugation, and the settling rates and directly measured yield stresses, using the vane method, were determined for every concentration of coal–water mixtures (CWM) used. The shear-stress/shear-rate dependences of the test suspensions, covering the shear-rate range from 0.1 to 105 s−1, were determined using a capillary rheometer. For some of the test suspensions, correction for a pronounced wall-slip effect was required. Apart from the directly measured yield stresses using the vane method, yield stresses were estimated indirectly by extrapolation and rheological model-fitting. Extrapolation of directly measured yield stresses to infinite limit was used to estimate maximum packing for comparison with those determined from the irreducible sediment volumes using centrifugation. The two-parameter power-law, Bingham plastic and Casson empirical rheological models, and the three-parameter Herschel-Bulkley and Sisko models were used to fit the shear-stress/shear-rate data. In general, the shear-stress/shear-rate dependence was found to be shear-thinning, power-law, over the lower ranges of shear, and to tend to Newtonian limit at high shear; a dependence which is best described by the three-parameter Sisko model.

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.

The properties, settling behavior and the rheology of fine particulate slurries were investigated. The test slurries used in these studies included laterite, gypsum, titanium dioxide and silica flour. All were industrial slurries. The property characterization experiments included measurements of the particle size distributions, particle surface areas, pore size distributions and particle densities. Slurries containing different concentrations of the test solids were made up, and their pH and zeta potentials were measured. The rates of settling of the slurries as a function of concentration were determined. The rheological data included directly measured yield stresses using the vane apparatus, and the shear stress-shear rate dependence over ranges of shear rate from less than 1 to more than 10 4 s −1 using both rotational and capillary rheometers. The settling rate data were analyzed using hindered settling correlations. The yield stress-concentration data were correlated using relationships based on interparticle spacing and the maximum packing volume fraction of solids. Both two-parameter as well as three-parameter empirical rheological models were used to fit the shear stress-shear rate data. The three-parameter Sisko model, which combines low- and intermediate-shear power-law with high-shear Newtonian limiting behavior, was found to provide the best overall description of the flow curves for all slurries, at all solids loadings, over the entire measured range of shear rates. This finding is significant because this model evidently works for most non-Newtonian solid-liquid suspensions, and also because it is applicable over precisely the range of shear rates relevant to pipeline transport of the suspensions.

The yield stresses of coal-water mixtures as functions of solids volume fraction, suspended particle size, and suspension pH were measured using the vane method. The suspension yield stress-solids volume fraction data were used to estimate the asymptotic values of solids concentration in the limit of infinite yield stress. The limiting solids volume fractions, obtained by such an extrapolation, were found to be approximately the same as those estimated independently from the irreducible sediment volumes obtained by centrifugation. The yield stress-concentration dependence was found to depend on particle size as well as pH, or the corresponding zeta potential which was also measured. However, for a given particle size mix the yield stress-concentration plots for different pH appeared to converge, as the yield stress became large, and to approach a common asymptotic value of solids concentration in the limit of infinite yield stress. A number of different extrapolation procedures were tested to establish the type of plot capable of providing a consistent, unambiguous value of these limiting concentrations. The procedures for estimating the maximum packing solids concentrations from centrifugal sedimentation were also tested by varying the speed and the duration of centrifugation. The solids concentration dependence of the yield stress of CWM was found to be described by two-parameter non-linear relationships based upon the mean interparticle spacing, and incorporating the maximum packing volume of solids. Furthermore, it is established that the yield stress is proportional to the reciprocal of the mean particle size.

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}.

The thermal conductivities of suspensions of particulate coals in water, in fuel oil and in other liquids, and of mixtures of different particulate solids in liquids were measured. The binary suspension data were used to determine the solid thermal conductivities of the particulate inclusions by various extrapolations, and to test the different extrapolation schemes. A general, second-order, effective conductivity formula for binary mixtures, valid for small differences between the conductivities of dispersed and continuous phases, was derived. For suspensions containing a mixture of N different solids, of solid to liquid conductivity ratios β t = k i k 0 and volume fractions φ i , the formulae ( k ∗ k 0 ) = β 1 φ1 β 2 φ2 … β N φN and ( k ∗ k 0 ) = [1 + ∑(β i 1 3 − 1)φ i ] 3 were derived for the suspension thermal conductivity k ∗ . The former was found to do a good job of prediction for binary and multi-solid suspensions. As the volume fraction weighted geometric mean of the constituent conductivities, this formula is known to lie between the harmonic and arithmetic means as lower and upper bounds, respectively. Furthermore, over the range of conductivity ratios β i where it works well, it either lies between tighter bounds based on variational principles, or when it falls outside it differs indistinguishably from these bounds, which in this case turn out to be quite close.

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 theoretical analysis of a dispersed flow biological reactor is presented in which Monod growth kinetics are used to describe the substrate utilization rate and the bacterial growth rate. Different bacterial and substrate dispersion coefficients are taken into account. Approximate perturbation solutions are established for two limiting situations, when the inverse Peclet number ϵ is very small and when it is very large. In the former case the perturbation analysis is singular. These perturbation solutions are compared with the solutions of the numerically integrated equations over a range of parameters corresponding to actual activated sludge systems. The application to the activated sludge system is discussed. On presente une analyse theorique d'un reacteur biologique a debit disperse et dans lequel on emploie la cinetique de croissance de Monod pour decrire le taux d'utilisation du substrat et celui de la croissance bacterienne. On considere differents coefficients de dispersion des bacteries et du substrat. On a etabli des solutions approximatives des perturbations dans le cas de deux situations limitatives, a savoir lorsque le nombre inverse de Peclet(ϵ) est tres petit et lorsqu'il est tres considerable; dans le premier cas, l'analyse des perturbationes est singuliere. On compare les solutions des perturbations avec celles d'equations integrees numeriquement, et ce, pour une echelle de parametres qui correspond a des systemes reels de boue activee. On discute l'application des resultats au systeme de boue activee.

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 detailed examination of the critical velocity in pipeline flow of non-colloidal slurries was carried out. Published critical velocity correlations were collected and were recast into a standard form so that they could be compared with each other, and they were also tested against a broad collection of experimental critical velocity data. Altogether, a total collection of 864 experimental critical velocity data, representing a broad variety of solid materials and pertaining to wide ranges of the variables involved, were used in these tests. This rather substantial body of experimental data was also used as the basis for developing a set of improved critical velocity correlations, which were established by fitting the data to various forms of the standard equation. Among the principal results of this work are the following: The dependence of the critical velocity on pipe diameter is very nearly equal to D 1 2 , while its dependence on particle size, for slurries of non-colloidal particles, is very nearly equal to d0. The analytical result due to Oroskar and Turian indicates that υc depends on pipe diameter as D0.6, and on particle size as d0, while the best empirical fits to the data suggest that the dependence on pipe diameter is approximately D0.5, and that on particle size is at most d0.06. In addition, both the newly established empirical correlations and the analytical correlation due to Oroskar and Turian predict a maximum in the υc vs. C curve, the maximum occurring at solids volume concentration of 0.25 to 0.30. The comparisons with experimental data further established that the analytical result by Oroskar and Turian, and the empirical correlations developed in this work, do a superior overall job of predicting the data than other published correlations.

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.

Resistance coefficients for flow of suspensions of well defined glass beads of narrow size fractions in 1-inch and 2-inch straight pipes, in standard 45°, 90° and 180° bends, in 90° smooth bends of various curvature radii, and also in gate and globe valves were measured. The measurements were made for two sizes of fine glass beads, -325 mesh and -200-325 mesh, covering wide ranges of turbulent Reynolds number and solids concentrations from 0 to 50 weight percent. These friction loss data were analyzed with regard to the effects of Reynolds number and suspended solids concentration, and the calculated resistance coefficients were compared with those estimated from available design procedures recommended for turbulent single-phase Newtonian flow. Within the range of particle sizes examined in this work no particle size effects could be discerned. The effects of Reynolds number and suspended solids concentration on the friction loss measurement were calculated.

Formal approximate solutions, for small ϵi, for the system where G is a general function of the N-dependent variables fi, are obtained by the method of matched asymptotic expansions. Uniformly valid approximations are composed for the region 0 ⩽ x ⩽ 1 by synthesizing expansions pertaining to three separate regions; near x = 0, near x = 1, and the intermediate region in between. Restrictions on the approximations are developed, and these can be used to evaluate the effects of the magnitudes of the λi, of the behavior of G, and of possible disparities in the values of the various parameters ϵi. The results obtained are of considerable practical value because they are readily applicable to actual situations, as demonstrated by a number of specific examples, and because it is possible to evaluate how good the approximation is in each individual case.

The plane Couette flow of a frictionally heated temperature dependent Ellis fluid is solved. The critical stress, above which steady-state temperature and velocity fields under prescribed boundary conditions cannot be established, is determined as a function of the Ellis number El and the parameter α.

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.

The temperature rise resulting from viscous dissipation in the fluid in a cone-and-plate viscometer is derived approximately using a variational method. In the derivation it is not necessary to make reference to any specific rheological model. It is concluded that for one commercially available instrument the maximum temperature rise is about 3°C. Hence viscous heating effects can lead to observable errors when a cone-and-plate instrument is operating at high speeds; such effects have been observed experimentally.

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.

Analytical solutions are given for the velocity and temperature profiles in a Newtonian fluid between two infinite parallel planes, one of which is moving fast enough that viscous dissipation heating effects are important. The temperature dependence of the viscosity and the thermal conductivity are taken into account. In one set of solutions both plates are maintained at the same temperature, and in another set of solutions there is zero heat flux through the moving plate. These results are then applied to the cone-and-plate viscometer, to show how the torque vs. angular speed curves deviate from linear behaviour because of the temperature dependence of the transport properties. The analytical expressions are compared with experimental flow curves, which seem to indicate that the viscous heating effects are slightly larger than the theoretical expressions would predict.

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.