Laminar flow of some inelastic time-dependent fluids in pipes

The paper presents some modification of the previously published model of thixotropic fluids for the case of fluids with irreversible destruction of the structure. In the second part of the paper the laminar flow of this kind of fluids in pipes is discussed. A simple method of determining the initial value of the structural parameter at the inlet to the pipe is proposed. D pipe diameter, m - k ₁,k ₂ rheological parameter in eq. (5), Nsⁿ/m² - L pipe length, m - m ₁,m ₂ rheological parameter in eq. (5) - n ₁,n ₂ rheological parameter in eq. (6) - p pressure, Pa - v mean linear velocity in the pipe, m/s - rheological parameter in eq. (2), = 1 s - shear rate, s^–1 - structural parameter - ₀ initial value of structural parameter - ₁, ₂ mean value of natural time for breakdown and build-up of the structure, respectively, s - shear stress, Pa - fluid density, kg/m³ - mean value of the friction factor - De modified Deborah number defined by eq. (11) - Re _m generalized Reynolds number defined by eq. (16) - Re _n generalized Reynolds number defined by eq. (10) - Se structural numbers defined eq. (12)     

Determination of turbulent pressure loss of non-newtonian oil flow in rough pipes

Summary A detailed theoretical analysis is carried out of the turbulent flow of non-newtonian fluids in rough pipes in which approximate assumptions are of negligible importance. A new friction-factor equation (BNS-equation) is obtained which is valid for turbulent flow of power-law type non-newtonian fluids in the transition region between smooth and wholly rough wall turbulence.The accuracy of the deduced formula was checked by in situ measurements on a crude oil pipeline having a diameter of 305 mm and a length of 161 km. For this pipeline the Reynolds numbers ranged between 10⁴ and 10⁵. The computed and the measured friction factors showed good agreement. The mean of the absolute values of the relative error was 4.1%, while the standard deviation was 0.81%, these results appear to give better fitting than other similar equations published earlier.The BNS-equation is the first analytically deduced relationship for non-newtonian crude oils which includes the effects of both Reynolds number and relative roughness of the pipe.With 6 figures and 2 tables     

An improved semi-empirical friction model for gas-liquid two-phase flow in horizontal and near horizontal pipes

Pressure drop and liquid hold-up are two very important fluid flow parameters in design and control of multiphase flow pipelines. Friction factors play an important role in the accurate calculation of pressure drop. Various empirical and semi-empirical closure relations exist in the literature to calculate the liquid-wall, gas-wall and interfacial friction in two-phase pipe flow.However most of them are empirical correlations found under special experimental conditions. In this paper by modification of a friction model available in the literature, an improved semiempirical model is proposed. The proposed model is incorporated in the two-fluid correlations under equilibrium conditions and solved. Pressure gradient and velocity profiles are validated against experimental data. Using the improved model, the pressure gradient deviation from experiments diminishes by about 3%; the no-slip condition at the interface is satisfied and the velocity profile is predicted in better agreement with the experimental data.     

Periodic flows of pseudoplastic fluids in pipes

Experimental results on the flow enhancement during flow of clay slurries (n = 0.15) through oscillating pipes are compared with theoretical predictions. The agreement is fairly good, especially in the oscillating boundary layer flow regime. Flow enhancement of order 10⁸ was found with the slurries used.     

Laser Doppler anemometry measurements of laminar flow in helical pipes

Three-dimensional laser Doppler anemometry measurements are performed on developed laminar flow in three helical pipes. The experimental observations are compared to results of numerical calculations employing the fully elliptic numerical method. Good agreement is found between measured data and numerical results. The three helical pipes, with curvature ratios of 0.0734 and 0.1374 and non-dimensional pitches of 0.0793 and 0.193, are adopted to study the effects of curvature and pitch on laminar flow in the experimental approach. The range of Reynolds numbers is 500–2000 to ensure laminar flow in the entire helical pipe. Both the profile shapes of the normal components of the secondary flow and those of the axial flow along the same centerline present not only similar patterns but also similar change when pitch, curvature ratio, and Reynolds number vary. The results demonstrate comprehensive relationships between the axial flow and the secondary flow.     

Analytical solution of flow of second-order non-Newtonian fluids through annular pipes

This paper presents an analytical solution to the unsteady flow of the second-order non-Newtonian fluids by the use of intergral transformation method.Based on the numerical results,the effect of non-Newtonian coefficient Hc and other parameters on the flow are analysed.It is shown that the annular flow has a shorter characteristic time than the general pipe flow while the correspondent velocity,average velocity have a(?)aller value for a given Hc.Else,when radii ratio keeps unchanged,the shear stress of inner wall of annular flow will change with the inner radius compared with the general pipe flow and is always smaller than that of the outer wall.     

Application of the Casson model to thixotropic waxy crude oil

The rheological behavior of a waxy crude oil was investigated using a coaxial cylinder viscometer. Experimental flow curves were fitted with the Casson equation. The Casson model was modified to interpret the hysteresis between upward and downward curves obtained in a series of consecutive runs. At the same shear rate, the mean axial ratio of the flow unit related to a down-curve (↓) is smaller than that related to an equilibrium up-curve (↑). This results in a decrease of the Casson yield stress τ_c↓ with respect to τ_c↑. A certain ξ coefficient describing departure from the equilibrium mean axial ratio was introduced into the model. Values of ξ calculated from the Casson yield stresses agreed satisfactorily with the theoretically predicted ones. Deviations from the Casson model at low shear rates were also explained. Received: 28 July 1998 Accepted: 18 February 1999     

Application of a borescope to studies of gas–liquid flow in downward inclined pipes

The aim of the present study is to investigate stratified downward gas–liquid pipe flow with a non-intrusive measurement technique that is based on a borescope connected to a digital video camera. The borescope-based technique enables to determine the instantaneous cross-sectional distribution of both phases within the pipe. Water and air were used as working fluids. Quantitative data was extracted from sequences of recorded video images by applying a developed data processing technique for instantaneous gas–liquid interface boundaries determination. Experiments were performed for a wide range of downward pipe inclinations and gas and liquid flow rates. The instantaneous and time-average cross-sectional holdup for each set of flow parameters was calculated. Particular attention was given to the study of the interface shape that in many occasions was not flat and was characterized by the penetration of the liquid along the pipe periphery. Temporal variation of the surface elevation was also studied and various regimes characterizing interfacial waves were defined using both the recorded time series of the instantaneous depth of the water layer and the Fourier analysis of those records.     

Pulsatile pipe flow of pseudoplastic fluids

This note is concerned with a laminar pipe flow of a non-Newtonian fluid under the action of a small pulsating pressure gradient superposed to a steady one. The constitutive law describing the rheological behaviour of the fluid is the so-called power law (Ostwald–de Waele). An approximated analytical solution is found for the velocity, as power series of the amplitude of the periodic disturbance. The analytic solution is compared with a direct numerical solution and the perfect accord of the values obtained is underscored.     

Combined free and forced convection for developed flow in curved pipes with finite curvature ratio

Combined free and forced convection for developed flow in a curved pipe with arbitrary curvature ratio is studied numerically. The curved pipe is heated with axially uniform heat flux, while the wall temperature is maintained peripherally uniform. The buoyancy force is accounted by the Boussinesq approximation. The effects of the Dean, Prandtl, and Rayleigh numbers and especially of a wide range of curvature ratios on the flow resistance and the average heat transfer rate are presented. The significant distortion of the dividing streamline and the appearance of the secondary flow with one dominant cell for pipe flow with higher buoyancy force and curvature ratio are also discussed.     

Nonlinear dynamics of extensible viscoelastic cantilevered pipes conveying pulsatile flow with an end nozzle

Nonlinear dynamics of an extensible cantilevered pipe conveying pulsating flow is considered in this paper. The fluid flow fluctuates harmonically and exhausts via a nozzle attached to the end of the pipe. Taking into account the extensibility assumption, the coupled nonlinear lateral–longitudinal equations of motion are derived using Hamilton's principle and discretized via Galerkin's method. The adaptive time step Adams algorithm is applied to extract the time response, and then the bifurcation, power spectral density and phase plane maps are plotted for some case studies. Effects of some geometrical parameters such as flow mass, pulsating flow frequency, gravity, nozzle mass and nozzle aspect ratio parameters are studied on the dynamics of such system and the validity of extensibility assumption is investigated and some conclusions are drawn.     

Transient effects of orthogonal pipe oscillations on laminar developing incompressible flow

This paper presents a numerical study of the transient developing laminar flow of a Newtonian incompressible fluid in a straight horizontal pipe oscillating around the vertical diameter at its entrance. The flow field is influenced by the tangential and Coriolis forces, which depend on the through‐flow Reynolds number, the oscillation Reynolds number and the angular amplitude of the pipe oscillation. The impulsive start of the latter generates a transient pulsating flow, whose duration increases with axial distance. In any cross‐section, this flow consists of a pair of symmetrical counter‐rotating vortices, which are alternatively clockwise and anti‐clockwise. The circumferentially averaged friction factor and the axial pressure gradient fluctuate with time and are always larger than the corresponding values for a stationary pipe. On the other hand, local axial velocities and local wall shear stress can be smaller than the corresponding stationary pipe values during some part of the pipe oscillation. The fluctuation amplitude of these local variables increases with axial distance and can be as high as 50% of the corresponding stationary pipe value, even at short distances from the pipe entrance. Eventually, the flow field reaches a periodic regime that depends only on the axial position. The results show that the transient flow field depends on the pipe oscillation pattern (initial position and/or direction of initial movement). Copyright © 2000 John Wiley & Sons, Ltd.     

Hyperbolicity and change of type in the flow viscoelastic fluids through pipes

We consider the steady flow of an upper convected Maxwell fluid through a pipe with wavy walls. The analysis is an extension to round pipes of the methods introduced by Yoo and Joseph [1] to study the same problem in plane channels. As in the channel problem, the vorticity in a small cylinder at the center of the pipe becomes hyperbolic when the centerline velocity is larger than the speed of shear waves into rest. The region of hyperbolicity is smaller, but the decay of vorticity is less, when the elasticity parameter is larger.     

Effects of buoyancy on heat transfer during turbulent flow of drag reducing fluids in vertical pipes

The effect of buoyancy on the heat transfer during upward turbulent flow of drag reducing fluids in vertical tubes has been theoretically analyzed. A criteria has been established for limiting the decrease in heat transfer to less than 5% for fluids of varying drag reducing ability. The final expression for quantitative evaluation of natural convection effect on forced convection could be applied to upward as well as downward turbulent flow of drag-reducing fluids merely by a change in sign of the controlling term.

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Einfluß des Auftriebes auf den Wärmeübergang bei turbulenter Strömung mit widerstandsreduzierenden Fluiden in vertikalen Rohren

Zusammenfassung Es wird der Einfluß des Auftriebes auf den Wärmeübergang bei turbulenter Aufwärtsströmung von widerstandsreduzierenden Fluiden in vertikalen Rohren theoretisch analysiert. Dabei wird ein Kriterium für die Begrenzung der Abnahme des Wärmeübergangs auf weniger als 5% für Fluide unterschiedlicher Widerstandsverringerung aufgestellt. Die endgültige mathematische Formulierung für die quantitative Beschreibung des Einflusses der natürlichen Konvektion auf die erzwungene Konvektion konnte sowohl auf Aufwärts-als auch auf Abwärtsströmung eines Fluids mit vermindertem Strömungswiderstand einfach dadurch angewandt werden, daß man ein Vorzeichen in dem die Strömung bestimmenden Term ändert.

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Nomenclature C _p specific heat at constant pressure (KJ/kg) - D pipe diameter (m) - De Deborah number - f friction factor - f reduced friction factor - g acceleration due to gravity (m/sec²) - Gr Grashof number - k thermal conductivity (KW/m °K) - Nu Nusselt number - Nu reduced Nusselt number - Pr Prandtl number - R pipe radius (m) - Re Reynolds number - Re reduced Reynolds number - T temperature (°C or °K) - T _b temperature of bulk of the fluid (°C or °K) - T _i initial temperature (°C or °K) - T _w wall temperature (°C or °K) - u ^* friction velocity (m/sec) - u ⁺ dimensionless axial velocity - V _m average velocity (m/sec) - y ⁺ dimensionless distance from the wall Greek symbols , functions ofDe (Table 1) - wall shear rate (sec^–1) - _fl fluid relaxation time (sec) - kinematic viscosity (m²/sec) - _B buoyant boundary layer thickness (m) - _m thickness of boundary sub-layer plus buffer layer (m) - _m ⁺ dimensionless thickness of boundary sub-layer plus buffer layer - _t thermal boundary layer (m) - density (kg/m³) - integrated density (kg/m³) - _b density of bulk of the fluid (kg/m³) - _w density of fluid at the wall (kg/m³) - _b viscosity of bulk of the fluid (Pa · sec) - _w viscosity of fluid at the wall (Pa · sec) - _w wall shear stress (N/m²) - _w reduced wall shear stress (N/m²) - change of shear stress across buoyant layer (N/m²)     

3D simulations of the flow of thixotropic fluids,in large-gap Couette and vane-cup geometries

Models of the vane-cup and Couette rheometers are compared using computational fluid dynamics as well as approximate solutions. Thixotropy of the fluid is incorporated by means of a model based on experimental data for various toothpastes. Parameters of the model are calculated by fitting the results of step-shear tests in Couette geometry and are subsequently used to predict torque for similar tests in vane-cup geometry. Calculations for the thixotropic model were compared to the calculations for a time-independent equilibrium model with the same steady-shear rheology. There are two important findings in this work. First, at constant angular velocity, torque in the vane-cup rheometer for the thixotropic model turns out greater than for the equilibrium model, because the structure of the thixotropic fluid has not had enough time to reach equilibrium. This has implications for rheometry and the modeling of mixing operations. In both cases, torques may be underestimated if the standard equilibrium model is used in calculations. The second finding is relevant to rheometry. As is well known, equilibrium model predicts a lower torque value for a vane-cup geometry than for an equivalent Couette geometry. We found that taking thixotropy into account either makes the difference less pronounced or in some cases actually makes torque for vane-cup higher than for Couette. End effects are also analyzed.     

An extended study of the hydrodynamics of gravity-driven film flow of power-law fluids

A new similarity transformation has been devised for extensive studies of accelerating non-Newtonian film flow. The partial differential equations governing the hydrodynamics of the flow of a power-law fluid down along an inclined plane surface are transformed into a set of two ordinary differential equations by means of the dimensionless velocity component approach. Although the analysis is applicable for any angle of inclination (0<π/2), the resulting one-parameter problem involves only the power-law index n. Nevertheless, physically essential quantities, like the velocity components and the skin-friction coefficient, do depend on and relevant relationships are deduced between the vertical and inclined cases. Accurate numerical similarity solutions are provided for n in the range from 0.1 to 2.0. The present method enables solutions to be obtained also for highly pseudo-plastic films, i.e. for n below 0.5. The mass flow rate entrained into the momentum boundary layer from the inviscid freestream is expressed in terms of a dimensionless mass flux parameter Φ, which depends on the dimensionless boundary layer thickness and the velocity components at the edge of the viscous boundary layer. Φ, which is thus an integral part of the similarity solution, turns out to decrease monotonically with n. This parameter is of particular relevance in the determination of the streamwise position at which the entire freestream has been entrained and viscous stresses prevail all the way to the free surface of the film. A short-cut method to facilitate rapid and yet accurate estimates of the mass flux parameter is developed to this end.     

A three-layer model for solid-liquid flow in horizontal pipes

A three-layer model for solid-liquid flow in horizontal pipes is proposed. This model overcomes the limitations of the two-layer model. The model predictions exhibit satisfactory agreement with the experimental data and existing correlations.     

Numerical solutions of pulsating flow and heat transfer characteristics in a pipe

Numerical studies are made of flow and heat transfer characteristics of a pulsating flow in a pipe. Complete time-dependent laminar boundary-layer equations are solved numerically over broad ranges of the parameter spaces, i.e., the frequency parameter β and the amplitude of oscillation A. Recently developed numerical solution procedures for unsteady boundary-layer equations are utilized. The capabilities of the present numerical model are satisfactorily tested by comparing the instantaenous axial velocities with the existing data in various parameters. The time-mean axial velocity profiles are substantially unaffected by the changes in β and A. For high frequencies, the prominent effect of pulsations is felt principally in a thin layer near the solid wall. Skin friction is generally greateer than that of a steady flow. The influence of oscillation on skin friction is appreciable both in terms of magnitude and phase relation. Numerical results for temperature are analyzed to reveal significant heat transfer characteristics. In the downstream fully established region, the Nusselt number either increases or decreases over the steady-flow value, depending on the frequency parameter, although the deviations from the steady values are rather small in magnitude for the parameter ranges computed. The Nusselt number trend is amplified as A increases and when the Prandtl number is low below unity. These heat transfer characteristics are qualitatively consistent with previous theoretical predictions.