Explicit calculation of the friction factor in pipeline flow of Bingham plastic fluids: a neural network approach

An artificial neural network (ANN) approach was used in this paper to develop an explicit procedure for calculating the friction factor, f, under both laminar and turbulent flow conditions of Bingham plastic fluids in closed conduits and pipe networks. The procedure aims at reducing the computational efforts as well as eliminating the need for conducting complex and time-consuming iterative solutions of the governing implicit equations for calculating the friction factor, f. The ANN approach involved the establishment of an explicit relationship among the Reynolds number, Re, Hedstrom number, He, and the friction factor, f, under both laminar and turbulent flow conditions. Although, an analytical solution of the governing equation under the laminar flow regime was also feasible (such an equation is also provided in this paper), the ANN model is applicable under both laminar and turbulent flow conditions where the analytical approach will have major limitations (especially when considering the implicit equation that govern the turbulent flow regime).     

Kinetic energy and momentum factors for rough pipes

Kinetic energy and momentum factors are derived, evaluated and presented for fully developed turbulent flow in both uniformly sand-roughened and commercial pipes as a function of Reynolds number and relative roughness. For the uniform sand roughness, the universal velocity profile and friction factor equations of Nedderman and Shearer were used, while for the commercial roughness the corresponding equations of Colebrook and White were employed.     

Analytical solutions for the Colebrook and White equation and for pressure drop in ideal gas flow in pipes

The friction factor, evaluated from the Colebrook and White equation, is traditionally computed iteratively. An analytical solution of the Colebrook and White equation for the friction factor can be obtained, using the Lambert W function. Also, the equation relating the outlet pressure to the inlet pressure of an ideal gas flowing through a straight pipe under isothermal, steady state conditions has been hitherto considered in literature to be implicit in these variables, probably due to its inherent non-linear nature. However, it can be shown that an analytical solution to the above equation for the pressure drop (or alternatively, outlet pressure) can also be obtained using the Lambert W function.     

A practical method for predicting the friction factor of power-law fluids in a rectangular duct

This article presents a simple and unique method for predicting the friction factor for the fully developed, laminar flow of power-law fluids in ducts with a rectangular cross-section by means of an engineering calculation for rapid equipment design. The relationship between friction factor f and the Reynolds number defined by Metzner–Reed (Re_MR) for a rectangular cross-section was investigated and rearranged with very good engineering accuracy using the results of very simple correlations based on the application of the well-known and long-established coefficient for Newtonian fluid flow. The investigated value of the flow index was up to 2, and the aspect ratio ranged from 0 to 1, with calculation error not higher than 3.5%. The proposed method was compared with conventional methods from the literature, and was validated by means of numerical computations, and also with experimental data from the literature. The product of friction factor and Re_MR is a linear function of geometrical parameter C and flow behavior index n and also the proposed method predicts the friction factor as accurately as conventional and traditional methods.     

Generalized correlation for friction loss in drag reducing polymer solutions

The definition of the pipe flow friction factor has been extended to include the effect of fluid viscoelastic properties on energy dissipation in turbulent tube flow. The resulting friction factor includes a characteristic fluid relaxation time, which can be determined directly from rheological measurements, and reduces to the usual Fanning friction factor for inelastic fluids. The use of this more general friction factor enables turbulent tube flow data for both fresh and shear degraded “concentrated” drag reducing polymer solutions of various concentrations in various tube sizes to be correlated by the usual f vs. N_Re relation for Newtonian fluids in smooth tubes.     

A new six parameter model to estimate the friction factor

A new explicit formula for estimating the friction factor using six parameters is proposed. The model was set up by considering the effect of residual stresses in the flow by two distinct contributions: the first is attributed to the flow velocity (Reynolds number) and the second to the duct roughness. Compared to other models, this new equation gives the best fit with Nikuradse's results. A new model to calculate the friction is proposed. The model is based on assuming the residual stresses due to the laminar to turbulent flow transition by two distinct contributions: the first is attributed to the flow velocity (Reynolds number) and the second to the duct roughness. Compared to other models, this new equation gives the best fit with respect of Nikuradse's results. The model does not consider the effect of pipe wall on the velocity distribution. © 2019 American Institute of Chemical Engineers AIChE J, 65: 1144–1148, 2019     

An explicit method of design of non-isothermal tube-wall reactor with volume change

An analysis of a tube-wall reactor with varying volume is considered under turbulent flow conditions for the case in which both the kinetic and diffusional resistances are important. The implicit molar flux expression is approximated by three successive approximations and the mass and energy balances are combined to obtain a single explicit design equation. This equation takes the form of a simple numerical integration which requires only physical and chemical properties of the inlet gas and reactor operating conditions. The method also predicts the temperature and heat flux along the reactor length. Comparison of the model predictions with a limited number of previous experimental results is good.     

An explicit equation for the terminal velocity of solid spheres falling in pseudoplastic liquids

An explicit equation is proposed which predicts directly the terminal velocity of solid spheres falling through stagnant pseudoplastic liquids from the knowledge of the physical properties of the spheres and of the surrounding liquid. The equation is a generalization of the equation proposed for Newtonian liquids. By properly defining the dimensionless diameter, d_*, a function of the Archimedes number, Ar, and the dimensionless velocity, U_*, a function of the generalized Reynolds number, Re, to account for the non-Newtonian characteristics of the liquid, the final equation relating these two variables has similar form to the Newtonian equation. The predictions are very good when they are compared to 55 pairs of Re—C_D for non-Newtonian data and 37 pairs for Newtonian data published previously. The root mean square error on the dimensionless velocity is 0.081 and much better than the only other equation previously proposed.     

Terminal settling velocity and drag coefficient of biofilm‐coated particles at high Reynolds numbers

The drag force (F_d) on bio‐coated particles taken from two laboratory‐scale liquid–solid circulating fluidized bed bioreactors (LSCFBBR) was studied. The terminal velocities (u_t) and Reynolds numbers (Re_t) of particles observed were higher than reported in the literature. Literature equations for determining u_t were found inadequate to predict drag coefficient (C_d) in Re_t > 130. A new equation for determining F_d as an explicit function of terminal settling velocity was generated based on Archimedes numbers (Ar) of the biofilm‐coated particle. The proposed equation adequately predicted the terminal settling velocity of other literature data at lower Re_t of less than 130, with an accuracy >85%. © 2010 American Institute of Chemical Engineers AIChE J, 2010     

Determination of the Scalar Friction Factor for Nonspherical Particles and Aggregates Across the Entire Knudsen Number Range by Direct Simulation Monte Carlo (DSMC)

The friction factor of an aerosol particle depends upon the Knudsen number (Kn), as gas molecule–particle momentum transfer occurs in the transition regime. For spheres, the friction factor can be calculated using the Stokes–Millikan equation (with the slip correction factor). However, a suitable friction factor relationship remains sought-after for nonspherical particles. We use direct simulation Monte Carlo (DSMC) to evaluate an algebraic expression for the transition regime friction factor that is intended for application to arbitrarily shaped particles. The tested friction factor expression is derived from dimensional analysis and is analogous to Dahneke's adjusted sphere expression. In applying this expression to nonspherical objects, we argue for the use of two previously developed drag approximations in the continuum (Kn → 0) and free molecular (Kn → ∞) regimes: the Hubbard–Douglas approximation and the projected area (PA) approximation, respectively. These approximations lead to two calculable geometric parameters for any particle: the Smoluchowski radius, R _S, and the projected area, PA. Dimensional analysis reveals that Kn should be calculated with PA/πR _S as the normalizing length scale, and with Kn defined in this manner, traditional relationships for the slip correction factor should apply for arbitrarily shaped particles. Furthermore, with this expression, Kn-dependent parameters, such as the dynamic shape factor, are readily calculable for nonspherical objects. DSMC calculations of the orientationally averaged drag on spheres and test aggregates (dimers, and open and dense 20-mers) in the range Kn = 0.05–10 provide strong support for the proposed method for friction factor calculation in the transition regime. Experimental measurements of the drag on aggregates composed of 2–5 primary particles further agree well with DSMC results, with differences of less than 10% typically between theoretical predictions, numerical calculations, and experimental measurements.

Copyright 2012 American Association for Aerosol Research     

Pressure drop during the flow of stokesian fluids through granular beds

The resistance to flow of Stokesian fluids (i.e. time—independent fluids with no yield stress) through granular beds is discussed. A definition of friction factor λ and generalized Reynolds number Re′_BK is proposed for fluids obeying the “power-law” shear stress—shear rate relation.The generalized Ergun equation, derived in this paper, gives the dependence of the friction factor on the generalized Reynolds number and flow behaviour index n. The validity of the generalized Ergun equation was proved experimentally. In the case of Newtonian fluid (for n = 1·0) a more exact form of the classical Ergun equation is obtained.     

A coupled CFD-Monte Carlo method for simulating complex aerosol dynamics in turbulent flows

A coupled computational fluid dynamics (CFD)-Monte Carlo method is presented to simulate complex aerosol dynamics in turbulent flows. A Lagrangian particle method-based probability density function (PDF) transport equation is formulated to solve the population balance equation (PBE) of aerosol particles. The formulated CFD-Monte Carlo method allows investigating the interaction between turbulence and aerosol dynamics and incorporating individual aerosol dynamic kernels as well as obtaining full particle size distribution (PSD). Several typical cases of aerosol dynamic processes including turbulent coagulation, nucleation and growth are studied and compared to the sectional method with excellent agreement. Coagulation in both laminar and turbulent flows is simulated and compared to demonstrate the effect of turbulence on aerosol dynamics. The effect of jet Reynolds (Re_j) number on aerosol dynamics in turbulent flows is fully investigated for each of the studied cases. The results demonstrate that Re_j number has significant impact on a single aerosol dynamic process (e.g., coagulation) and the simultaneous competitive aerosol dynamic processes in turbulent flows. This newly modified CFD-Monte Carlo/PDF method renders an efficient method for simulating complex aerosol dynamics in turbulent flows and provides a better insight into the interactions between turbulence and the full PSD of aerosol particles.

Copyright © 2017 American Association for Aerosol Research     

Fine particle deposition in laminar and turbulent flows

The deposition of fine silica and polystyrene spheres was measured for conditions of laminar and turbulent flow (960 ≤ Re ≤ 16040) in a rectangular channel using image analysis. The plate glass deposition surfaces were rendered positively charged by coating them with a cationic copolymer while, under the water chemistry conditions employed, both types of particles were negatively charged. It was found that, contrary to the results for laminar flow, the initial depositon rates in turbulent flow decreased with increasing Re, indicating that deposition was no longer mass-transfer controlled and that particle attachment played an increasingly important role as Re was raised. Attachment was modelled as a rate process in series with mass transfer in which the attachment rate varies inversely as the square of the friction velocity. Under the conditions of the present experiments, no particle re-entrainment was observed, so that the declining rate of particle accumulation on the wall recorded in each run could only be attributed to a declining deposition rate. Even where asymptotic accumulations were reached, particle coverages never exceeded 3.5%.     

Accuracy of approximations in kinetic data circulation

Kinetic data can be determined from the relation of two kinetic equations in their integrated form. For the calculation of the exponential integral, approximations are generally used, which allow a simple calculation of the activation energy. With respect of the most frequently used approximations, an evaluation of error produced by their application has been calculated. The best results can be achieved by using the equation which permits determination of E to within 1% accuracy.     

COMPARISON OF THE VELOCITIES AND THE WALL EFFECT BETWEEN SPHERES AND CUBES IN THE ACCELERATING REGION

The velocities and the wall effect for spheres and cubes in the accelerating region were compared experimentally for very high Reynolds number (18000 < N _Re  < 77000 for spheres and 12000 < N _Re  < 49000 for cubes). Experiments were conducted with 0.012, 0.015, 0.018, 0.021, 0.025, and 0.03175 m spheres and their respective cubes of equal volume falling in water inside cylinders of diameters equal to 0.034, 0.049, 0.07, 0.10, 0.14, and 0.19 m. The velocities of the spheres were always greater than the velocities for the respective cubes of equal volume. The wall effect increases with the particle size and fall distance. The relative wall effect between the spheres and the cubes varies with the fall distance and the ratio of the particle to tube diameter. The wall factor as a function of particle-to-tube ratio for the spheres is described well by the Munroe equation (5.7% absolute error), while for the cubes the points are more dispersed.     

Mixed convective boundary layer flow over a vertical wedge embedded in a porous medium saturated with a nanofluid: Natural Convection Dominated Regime

A boundary layer analysis is presented for the mixed convection past a vertical wedge in a porous medium saturated with a nano fluid. The governing partial differential equations are transformed into a set of non-similar equations and solved numerically by an efficient, implicit, iterative, finite-difference method. A parametric study illustrating the influence of various physical parameters is performed. Numerical results for the velocity, temperature, and nanoparticles volume fraction profiles, as well as the friction factor, surface heat and mass transfer rates have been presented for parametric variations of the buoyancy ratio parameter Nr, Brownian motion parameter Nb, thermophoresis parameter Nt, and Lewis number Le. The dependency of the friction factor, surface heat transfer rate (Nusselt number), and mass transfer rate (Sherwood number) on these parameters has been discussed.     

Combined forced and free convection flow past a horizontal flat plate

The problem of simultaneous forced and free convection flow of a Newtonian fluid past a hot or cold horizontal flat plate is investigated by means of numerical solutions of the full equations of motion and thermal energy subject only to the Boussinesq approximation. These solutions span the parameter ranges 10 ≤ Re ≤ 100, 0.1 ≤ Pr ≤ 10, and –2.215 ≤ Gr/Re^5/2 ≤ 2.215 where Re, Pr, and Gr are based on the ambient free stream fluid properties and the overall plate length l. When Gr > 0, the boundary flow near the plate surface is accelerated relative to the corresponding forced convection flow, with a resulting increase in both the local skin friction and heat transfer coefficients. When Gr < 0, the boundary flow is decelerated, the local skin friction and heat transfer are decreased, and the flow actually separates for Gr/Re^5/2 < –0.8 when Pr = 0.7. In the latter circumstance, an increasing degree of upstream influence is observed as Gr/Re^5/2 is further decreased.     

Pressure drop for single and two‐phase flow of non‐newtonian liquids in helical coils

New experimental results on pressure loss for the single and two‐phase gas‐liquid flow with non‐Newtonian liquids in helical coils are reported. For a constant value of the curvature ratio, the value of the helix angle of the coils is varied from 2.56° to 9.37°. For single phase flow, the effect of helix angle on pressure loss is found to be negligible in laminar flow regime but pressure loss increases with the increasing value of helix angle in turbulent flow conditions. On the other hand, for the two‐phase flow, the well‐known Lockhart‐Martinelli method correlates the present results for all values of helix angle (2.56‐9.37°) satisfactorily under turbulent/laminar and turbulent/turbulent conditions over the following ranges of variables as: 0.57 ≤ n′ ≤ 1; Re′ < 4000; Re_l < 4000; Re_g < 8000; 8 ≤ x ≤ 1000 and 0.2 ≤ De′ ≤ 1000.