Free coating of non-newtonian liquids onto a vertical surface

The coating of non-Newtonian liquids onto a vertical surface continuously withdrawn from the liquid bath is considered. An analytic treatment is presented for purely viscous non-Newtonial liquids using the Ellis and generalised Bingham models both of which may be reduced to a new theory for power-law fluids. The theories give a relationship between the dimensionless film thickness, T₁, and the Capillary number, C₁, as a function of the fluid physical properties and the parameters of the viscous model. The dimensionless groups have been generalised to allow for non-Newtonian behaviour. The power-law and Ellis model predictions are compared with previous theoretical studies and shown to be consistent with known limits. Experimental data are also presented for a wide range of non-Newtonian fluids and compared with the new theories.     

Trickle bed hydrodynamics and flow regime transition at elevated temperature for a Newtonian and a non-Newtonian liquid

Despite the hydrodynamics of trickle beds experiencing high pressures has become largely documented in the recent literature, trickle bed hydrodynamic behavior at elevated temperatures, on the contrary, largely remains terra incognita. This study's aim was to demonstrate experimentally the temperature shift of trickle-to-pulse flow regime transition, pulse velocity, two-phase pressure drop, liquid holdup and liquid axial dispersion coefficient. These parameters were determined for Newtonian (air-water) and non-Newtonian (air-0.25% Carboxymethylcellulose (CMC)) liquids, and the various experimental results were compared to available literature models and correlations for confrontation and recommendations. The trickle-to-pulse flow transition boundary shifted towards higher gas and liquid superficial velocities with increasingly temperatures, aligning with the findings on pressure effects which likewise were confirmed to broaden the trickle flow domain. The Larachi-Charpentier-Favier diagram [Larachi et al., 1993, The Canadian Journal of Chemical Engineering 71, 319-321] provided good predictions of the transition locus at elevated temperature for Newtonian liquids. Conversely, everything else being kept identical, increasingly temperatures occasioned a decrease in both two-phase pressure drop and liquid holdup; whereas pulse velocity was observed to increase with temperature. The Iliuta and Larachi slit model for non-Newtonian fluids [Iliuta and Larachi, 2002, Chemical Engineering Science 46, 1233-1246] predicted with very good accuracy both the pressure drops and the liquid holdups regardless of pressure and temperature without requiring any adjustable parameter. The Burghardt et al. [2004, Industrial and Engineering Chemistry Research 43, 4511-4521] pulse velocity correlation can be recommended for preliminary engineering calculations of pulse velocity at elevated temperature, pressure, Newtonian and non-Newtonian liquids. The liquid axial dispersion coefficient (D_ax) extracted from the axial dispersion RTD model revealed that temperatures did not affect in a substantial manner this parameter. Both Newtonian and power-law non-Newtonian fluids behaved qualitatively similarly regarding the effect of temperature.     

Gas—liauid mass transfer in three-phase stirred tank reagors: Newtonian and non-newtonian fluids

Gas—liquid mass transfer has been investigated in gas—liquid-solid three-phase stirred tank reactors with Newtonian and non-Newtonian liquids. Volumetric mass transfer coefficients and gas hold-ups were measured in a 0.2 m i.d. stirred tank reactor and the effects of low-density polymeric particles (ρ_s, =1030 and 1200 kg/m³; up to 15 vol%) on gas—liquid mass transfer were examined. The volumetric mass transfer coefficients in water were found to decrease due to the presence of solid particles at constant impeller speed and superficial gas velocity. On the other hand, solids loading led to higher mass transfer rates in non-Newtonian carboxymethyl cellulose aqueous solutions. Our previously proposed model for mass transfer in gas—liquid two-phase systems was extended to gas—liquid—solid three-phase systems. Reasonable agreement was found between the predictions of the proposed model and the experimental data.     

Resistance properties of coal-water paste flowing in pipes

On the basis of analyzing the mechanism of coal-water paste (CWP) slip flow, the similitude criterion, known as general Reynolds number Re_g which can characterize the state of non-Newtonian fluid flow in pipes, was put forward. The energy loss coefficient of CWP laminar flow has the same form as Newtonian fluid, i.e. λ=64/Re_g. Re_g holds good not only for the slip flow of non-Newtonian fluid, but also for slip-free flow as a steady-state laminar flow in pipes. The results obtained from experiments show that the energy loss coefficient of CWP laminar flow in pipes has the same form as Newtonian fluid.     

A general correlation for purely viscous non-newtonian fluids flowing in ducts of arbitrary cross-section

An alternative correlation for the flow of purely viscous non-Newtonian fluids in ducts of arbitrary cross-section is proposed. It involves two dimensionless groups which are Unique relationships exist between ? and Re^*_G for both laminar and turbulent flows which enable direct predictions of pressure drop from flow rate results or vice-versa. Excellent agreement between the new correlations and available experimental data for purely viscous non-Newtonian fluids flowing in circular tubes and square ducts is demonstrated.     

Mass transfer of carbon dioxide in aqueous polyacrylamide solution with methyldiethanolamine

Carbon dioxide was absorbed into aqueous polyacrylamide (PAA) solution containing methyl-diethanolamine (MDEA) in a flat-stirred vessel to investigate the effect of non-Newtonian rheological behavior of PAA on the rate of chemical absorption of CO₂, where the reaction between CO₂ and MDEA was assumed to be a first-order reaction with respect to the molar concentration of CO₂ and MDEA, respectively. The liquid-side mass transfer coefficient (k_L), which was obtained from the dimensionless empirical equation containing the viscoelasticity properties of a non-Newtonian liquid, was used to estimate the enhancement factor due to chemical reaction. PAA with elastic property of non-Newtonian liquid made the rate of chemical absorption of CO₂ accelerate compared with a Newtonian liquid     

Effect of pseudoplastic behaviour of liquid in co-current three-phase fluidized beds on bed expansion

Gas hold-up and bed expansion measurements were carried out for a bed of glass beads fluidized in Newtonian liquids and non-Newtonian liquids with gas. The value of gas hold-up increased and decreased with increasing particle size and liquid velocity, respectively. The effect of rheological properties on gas hold-up was insignificant and therefore the gas hold-up data for both Newtonian and non-Newtonian fluids were reasonably fitted by the available correlation which had no liquid viscosity term. The bed voidage increased with increasing superficial liquid velocities and superficial gas velocities. The increase of the viscous non-Newtonian flow behaviours resulted in an increase of the bed voidage. The correlation for the bed voidage in three-phase fluidized beds was developed for gas-Newtonian or non-Newtonian liquid–solid three-phase systems by combining the generalized wake model and the correlation for liquid–solid two-phase systems proposed previously by the authors. The predictions for bed voidage were in reasonable agreement with the present experimental data for three-phase systems with Newtonian and non-Newtonian liquids in a wide range of Reynolds numbers.     

Experimental and CFD investigation of convective heat transfer in helically coiled tube heat exchanger

Experimental studies on isothermal steady state and non-isothermal unsteady state conditions were carried out in helical coils for Newtonian as well as for non-Newtonian fluids. Water and glycerol–water mixture (10 and 20% glycerol) were used as Newtonian, and 0.5–1% (w/w) dilute aqueous polymer solutions of Sodium Carboxy Methyl Cellulose (SCMC) and Sodium Alginate (SA) as non-Newtonian fluids are used in this study. These experiments were performed for coil curvature ratios as δ = 0.0757, 0.064 and 0.055 in laminar and turbulent flow regimes (total 258 tests). The CFD analyses for laminar and turbulent flow were carried out using FLUENT 12.0.16 solver of CFD package. The CFD calculation results (Nu_i, U, T₂ and Tw_o) for laminar and turbulent flow are compared with the experimental results and the work of earlier investigators which were found to be in good agreement. For the first time, an innovative approach of correlating Nusselt number to dimensionless number, ‘M’, Prandtl number and coil curvature ratio using least-squares power law fit is presented in this paper which is not available in the literature. Several other correlations for calculation of Nusselt number for Newtonian and non-Newtonian fluids, and two correlations for friction factor in non-Newtonian fluids (based on 78 tests and 138 tests) are proposed. These developed correlations were compared with the work of earlier investigators and are found to be in good agreement.     

Inclination effects on wave characteristics in annular gas–liquid flows

Measurements of wave characteristics have been conducted in a 0.0762 m internal diameter (ID) pipe at inclinations of 0°, 10°, 20°, 45°, 60°, 75°, and 90° from horizontal. Wave celerity and frequency are very strongly dependent on modified Lockhart–Martinelli parameter, X*, and the inclination angle. Wave amplitude increases with increasing liquid film thickness at the bottom of the pipe. Wave amplitude depends on liquid film thickness for any pipe diameter, surface tension, and viscosity. Strouhal number (dimensionless wave frequency) decreases with increasing X*. Effect of pipe diameter, surface tension, and liquid viscosity on the liquid film Reynolds number, Re_LF, was studied. Re_LF variation with X* is not sensitive to the surface tension and less sensitive to the pipe diameter. However, Re_LF is very sensitive to the viscosity of the flowing liquid. Correlations for wave celerity, amplitude, frequency, and liquid film Reynolds number are proposed. © 2011 American Institute of Chemical Engineers AIChE J, 2012     

Isothermal and nonisothermal,laminar, inelastic,non-Newtonian tube-entrance flow following a contraction

The general equations of motion, including inertial and axial-diffusion terms, and the equation of energy, including axial-diffusion and radial-convection terms, were coupled with a temperature-dependent form of the Powell-Eyring flow equation and then solved simultaneously by numerical methods employing quasilinearization and the method of lines for flow from a larger tube of circular cross section through an abrupt contraction into a smaller tube. The small-tube walls were at a temperature equal to or higher or lower than that of the entering fluid. The Powell-Eyring equation has been demonstrated to represent inelastic flow better than do other simple equations for many simple non-Newtonian fluids, including suspensions of solids in fluids. Results are reported for the case in which the fluid does not adhere to the upstream-tube surface but does adhere to the smaller-tube surface, which approximates flow into the tubes of shell-and-tube heat exchangers and reactors. The computer results presented here included radial and axial velocity profiles; temperature profiles; and lengths of the small tube, L_eq, having a fully-developed-flow pressure loss equivalent to the excess pressure loss attributable to flow contraction and development in the tube. These are presented for a large-tube-small-tube-diameter ratio of 2 as functions of N_Pe of 5–100; N_Re of 0.01–100; z/D for the smaller tube; ψ(H), a temperature factor; and the Powell-Eyring-equation constants. In plots of L_eq against a normalized N_Re, the isothermal-flow non-Newtonian data appear to be close to computed Newtonian functions at higher N_Re. The effects of axial diffusion of momentum and energy and the conditions where they become important are quantatively shown. The results are in good agreement with previous reports in those circumstances where comparisons are appropriate.     

The Voidage Function for the Drag Force Ratio in a Liquid‐Fluidized Bed

Di Felice (1994) has shown that the ratio of the drag coefficient, C_D, on a sphere in a liquid‐fluidized bed of uniform spheres to the drag coefficient, C_DS, on the same sphere in isolation and subjected to the same superficial liquid velocity, u, is given by a function ?^?β, where β was expressed as an empirical function of the particle Reynolds number, Re = duρ/µ. Here it is shown that C_D/C_DS is well approximated by ?^?mm, where the Richardson‐Zaki index n is a function of the terminal free‐settling Reynolds number, Re_t = du_tρ/µ, and m is 2 plus the slope of the standard log C_DS vs. log Re plot at plot at Re = Re_t. The present model, using the best experimentally confirmed equation for n and a new simple equation for and a new simple equation for m, is compared with that of Di Felice in their respective abilities to predict liquid‐fluidized bed expansion.     

Pressure drop in two phase cocurrent upward flow in packed beds: Air/non-newtonian liquid systems

New data on the two phase pressure drop for the concurrent upflow of air-liquid (Newtonian and non-Newtonian) mixtures through packed beds of spherical and non-spherical particles are presented. The results for single phase flows and for the air-Newtonian liquid mixtures have been used both to gauge the overall accuracy of the present experimental methods and to evaluate the validity of the predictive expressions available in the literature. The two phase pressure drop has been measured as a function of the liquid and gas flow rates, column diameter and the power law model constants. Depending upon a suitable combination of the gas and liquid fluxes and the power law index, the two phase pressure drop may be less than its value for the flow of liquid alone. A simple expression is proposed which correlates the present set of experiments (nearly 500 data points) with satisfactory levels of accuracy over the following ranges of conditions: 0.54 ≤ n ≤ 1; 0.001 ≤ Re_L^* ≤ 50; 3.7 ≤ Re_G ≤ 177 and 0.9 ≤χ (Lockhart-Martinelli parameter) ≤ 104.     

Hindered settling in non-newtonian power law liquids

New experimental results on the hindered settling of model glass bead suspensions in non-Newtonian suspending media are reported. The data presented encompass the following ranges of variables: 7.38 × 10^?4 ≤ Re_1∞ ≤ 2; 0.0083 ≤ d/D ≤ 0.0703; 0.13 ≤ C ≤ 0.43 and 1 ≥ n ≥ 0.8. In these ranges of conditions, the dependence of the hindered settling velocity on concentration is adequately represented by the corresponding Newtonian expressions available in the literature. The influence of the power law flow behaviour index is completely embodied in the modified definition of the Reynolds number used for power law liquids.     

Mass transfer from ensembles of fluid spheres to a power-law liquid at moderate Reynolds and Peclet numbers

The steady convective mass transfer from ensembles of mono-size Newtonian fluid spheres to power-law liquids has been studied at moderate Reynolds and Peclet numbers. The species continuity equation segregated from momentum equations has been solved numerically using a finite difference method. A simple cell model has been used to account for the modification of the flow field due to the neighbouring droplets. Extensive numerical results have been obtained which elucidate effects of the Reynolds number (Re_o), Schmidt number (Sc), power-law index (n_o), internal to external fluid characteristic viscosity ratio (k) and the volume fraction of the dispersed phase (Φ) on the rate of mass transfer. The ranges of parameters considered herein are: 1?Re_o?200, 1?Sc?10000, 0.6?n_o?1.6, 0.1?k?50 and 0.2?Φ?0.6. For shear-thinning fluids (n_o<1), the rate of mass transfer is somewhat enhanced whereas for shear-thickening fluids (n_o>1), it decreased as compared to that in Newtonian fluids (n_o=1). A simple predictive correlation has been proposed which can be used to estimate the rate of mass transfer in liquid-liquid systems in a new application involving power-law continuous phase.     

Gas absorption with or without chemical reaction in laminar non-newtonian falling liquid films

An analytical solution is presented for gas absorption with or without a first-order or zero-order chemical reaction in a laminar non-Newtonian power-l model falling liquid film. For physical absorption, the first ten eigenvalues, series coefficients and related quantities are computed accurately by a quasinumerical method which shows considerable improvement over previous investigations. The range of applicability of the penetration theory solution is also established to indicate in what regions will the finite film thickness and complete velocity profile be important in determining the absorption rate. It is found that the range of dimensionless axial contact length X* in which the penetration theory is valid diminishes rapidly with increasi values of the power-law index n. For chemical absorption, the solution can be obtained by a linear superposition principle in terms of a “transie part” in which the effect of hydrodynamics within the liquid film is of importance and a “steady part” in which the reaction rate is controlling. In the “transient part” solution, the first ten eigenvalues and related quantities are reported for a variety of values of n and the dimensionl reaction rate parameter k_l* or k₀*. Certain asymptotic solutions from the penetration theory are also given and their range of applicab estimated. For any given n, it is estimated that only when k₁* or k₀* is less than approximately 10 will the finite film thickness and velocity profile have any effect on the absorption rate as compared to that calculated from the penetration theory with chemical reaction. The non-Newt character of the liquid film also has a significant influence on the absorption rate. At a fixed X*, the absorption enhancement due to reaction is when n = ∞ and is smallest when n = 0. The solutions obtained in this work are useful either for predicting absorption rates or for deter molecular diffusivity (and reaction rate constant) of gases in non-Newtonian falling liquid films.     

Non-Newtonian thin films: Theory and experiment

The flow of thin films of Newtonian and non-Newtonian, Power law fluids down a vertical plate was studied in the laminar and wavy flow regimes. A theoretical development based on the existence of a steady, periodic solution to the equations describing film flow was used to predict the flow characteristics of the film. Double logarithmic plots of film thickness versus Reynolds number (Re_{P. L.}) were linear, as predicted, up to a Re_{P. L.} = 100. The ratio of surface to average velocity was found to be approximately independent of the Reynolds number at values predicted by a laminar velocity profile. The wavelength of stable waves was found to be independent of the Reynolds number for a given fluid.     

Laminar mixing of shear thinning fluids in a SMX static mixer

Flow and mixing of power-law fluids in a standard SMX static mixer were simulated using computational fluid dynamics (CFD). Results showed that shear thinning reduces the ratio of pressure drop in the static mixer to pressure drop in empty tube as compared to Newtonian fluids. The correlations for pressure drop and friction factor were obtained at Re_MR?100. The friction factor is a function of both Reynolds number and power-law index. A proper apparent strain rate, area-weighted average strain rate on the solid surface in mixing section, was proposed to calculate pressure drop for a non-Newtonian fluid. Particle tracking showed that shear thinning fluids exhibit better mixing quality, lower pressure drop and higher mixing efficiency as compared to a Newtonian fluid in the SMX static mixer.     

Transition velocity and bed expansion of two-phase (liquid-solid) fluidization systems

Hydrodynamic transition experiments for two-phase (liquid-solid), both upward and downward, liquid flow systems were performed in a 127-mm diameter column. The particles were 3.2-mm polymer (1,280 kg/m³), 5.8mm polyethylene (910, 930, 946 kg/m³), 5.5-mm polystyrene (1,021 kg/m³) and 6.0-mm glass (2,230 kg/m³) spheres, with water, aqueous glycerol solution and silicone oil as liquids. The dimensionless pressure gradient increases initially with increasing liquid velocity, but decreases gradually with increasing liquid velocity beyond U_lmf due to bed expansion. The non-dimensionalized pressure gradient using the liquid/solid mixture density increases with increasing liquid velocity and then reaches a constant value close to unity beyond U_lmf. The minimum fluidization Reynolds number for liquid-solid system increases with increasing Archimedes number including both heavier and lighter than the density of the liquid phase. U_lmf should be the same for both upward and downward fluidization systems since the Ergun equation is based on the main assumption that drag force of the superficial liquid velocity, U_lmf, is equal to the net difference between gravitational and buoyancy forces.     

Dimensional analysis for planetary mixer: Mixing time and Reynolds numbers

Mixing time number is a convenient parameter to characterize mixing performance of stirred tanks. This dimensionless number is now well established for agitated vessels equipped with vertically and centrally mounted impeller for Newtonian as well as for non-Newtonian fluids. To our knowledge, there is more ambiguity concerning its definition for planetary mixers especially when they have dual motion (around two perpendicular axes) to achieve homogenization. In this study, dimensional analysis of mixing time and reliability of the modified Reynolds and mixing time numbers are proposed for such a planetary mixer particularly named as TRIAXE^® system. These two numbers are based on the maximum tip speed of mixer as the characteristic velocity. Modified dimensionless numbers are consistent with the definition of conventional Reynolds and mixing numbers (when only one revolving motion around the vertical axis of the mixing device occurs in the vessel).Mixing time experiments with TRIAXE^® mixer for highly viscous Newtonian fluids showed that the proposed modified Reynolds and mixing time numbers succeeded to obtain a unique mixing curve irrespective of the different speed ratio chosens. This agreement proves that the proposed modified dimensionless numbers can be well adapted for engineering purposes and they can be used to compare the mixing performance of planetary mixers.     

Drag coefficients of irregularly shaped particles

The steady-state free-fall conditions of isolated groups of ordered packed spheres moving through Newtonian fluids have been studied experimentally. Measurements of the drag coefficients are reported in this paper for six different geometrical shapes, including isometric, axisymmetric, orthotropic, plane and elongated conglomerates of spheres. From these measurements, a new and accurate empirical correlation for the drag coefficient, C_D, of variously shaped particles has been developed. This correlation has been formulated in terms of the Reynolds number based on the particle nominal diameter, Re, the ratio of the surface-equivalent-sphere to the nominal diameters, d_A/d_n, and the particle circularity, c. The predictions have been tested against both the experimental data for C_D collected in this study and the ones reported in previous works for cubes, rectangular parallelepipeds, tetrahedrons, cylinders and other shapes. A good agreement has been observed for the variously shaped agglomerates of spheres as well as for the regularly shape particles, over the ranges 0.15<Re<1500, 0.80<d_A/d_n<1.50 and 0.4<c<1.0.