Laminar Flow of Non-Newtonian Fluids in Concentric Annuli

1963 ◽  
Vol 3 (04) ◽  
pp. 274-276 ◽  
Author(s):  
Robert D. Vaughn
Keyword(s):  
Shear Stress ◽  
Shear Rate ◽  
Laminar Flow ◽  
Power Law ◽  
Experimental Work ◽  
Newtonian Fluids ◽  
Horizontal Line ◽  
Parallel Plates ◽  
Bingham Plastic ◽  
Fluid Behavior

The limiting cases of non-Newtonian fluids flowing inside a concentric annular duct are developed without using a model of the fluid behavior. The solutions provide limits with which to test the various models of fluid behavior such as the power law and Bingham plastic models. The results of previous theoretical work are discussed in terms of limiting cases. This limiting case study also shows that experimental work on flow of non-Newtonian fluids in annular ducts should be confined to ducts for which the ratio of the radius of the inner wall to that of the outer wall is less than 0.3 and preferably less than 0.2. Introduction During the last 10 years the problem of laminar flow of non-Newtonian fluids in concentric annuli has received much attention largely because of its application to the hydrodynamics of the wellbore. Recently solutions utilizing the power law and Bingham plastic models have been published.In this paper the method of limiting cases, which has been successfully applied to laminar-flow heat transfer will be applied to the problem of flow of non-time dependent, non-Newtonian fluids through annuli. This method permits solutions for the limiting cases to be made without using a model of unknown validity. The solutions, therefore, provide limits with which to test the various models which have been or will be proposed. A pertinent conclusion concerning the region of experimental work is also provided. DEVELOPMENT OF LIMITING CASES The limiting cases for the axial flow of fluids in concentric annuli may be defined with reference to Fig. L It is possible to define two limiting cases which pertain to the physical dimensions of the annulus. First, the annulus must degenerate to a circular pipe as the radius of the inner wall decreases or, as K = (KR/R) - 0. Second, the annulus must approach the limit of parallel plates of infinite extent as the spacing between the inner and outer tubes becomes small in comparison with the radius R of the annulus, or as K - 1. It is also possible to ascertain three limbing cases which pertain to fluid behavior. With reference to Fig. 2, as a fluid becomes progressively more pseudoplastic, the shear stress- shear rate relationship progressively approaches the indicated horizontal line more closely. At this limit the shear stress becomes independent of the shear rate. At the other extreme of increasingly dilatant behavior, the vertical asymptote is approached. Intermediate between these two limiting cases lies the case of the Newtonian fluid. Fluids which exhibit a yield shear stress also approach the limbing case of "infinite" pseudoplastic behavior. SPEJ P. 274^

Rheometric Tests and Extrusion

1951 ◽  
Vol 24 (3) ◽  
pp. 520-540
Author(s):  
Silvio Eccher
Keyword(s):  
Shear Stress ◽  
Shear Rate ◽  
Laminar Flow ◽  
Experimental Determination ◽  
Power Law ◽  
Reasonable Agreement ◽  
Straight Line ◽  
Flow Through ◽  
The Relationship

Abstract A cylindrical rheometer of the Couette type, suitable for the experimental determination of the rheological properties of extruded materials, was designed to provide data which could not be obtained with existing plastometers. The purpose of this study was strictly practical, as the work was performed in connection with a study of extruders. The results obtained on twenty-five different materials—natural and synthetic rubbers and compounds of both with various fillers—are reported; measurements fall within shear rate limits from 1 to 100 seconds−1. In this interval the relationship between logD (rate of shear) and logτ (shear stress) is nearly a straight line. It may, therefore, be analytically interpreted by the power law : D=−(τ/c)n, where n and c are parameters characteristic of the material. As the power law is known to be of limited validity, attempts were made to ascertain the limits of its application in laminar flow through a cylindrical hole. The results of measurements carried out on a 2-inch extruder and employing the same materials as were tested by the rheometer are reported. Measurements of pressure and flow were made, using discharge holes of various diameters and operating the screw at various speeds. Reasonable agreement was found between values of flow and pressure determined with an extruder and those calculated from parameters n and c determined with the cylindrical rheometer.

Dispersion of matter in non-Newtonian laminar flow through a circular tube

1966 ◽  
Vol 292 (1429) ◽  
pp. 203-208 ◽  
Keyword(s):  
Laminar Flow ◽  
Newtonian Fluid ◽  
Circular Tube ◽  
Newtonian Fluids ◽  
Parallel Plane ◽  
Bingham Plastic ◽  
Ellis Model ◽  
Model Fluid ◽  
Flow Through

Taylor’s analyses of the dispersion of Newtonian fluids in laminar flow in a circular tube are extended to the flow of the Bingham plastic and Ellis model fluid. The previous results for the Newtonian fluid and power-low fluid can be deduced from the results of this work. It is indicated that Aris’s modification of Taylor’s analyses can be naturally applied to the non-Newtonian fluid. Results obtained for laminar flow between two parallel plane walls are given in the appendix.

Influence of Pipe Rotation and Reciprocation in Flow of Non-Newtonian Fluids in Eccentric Annular Spaces

2006 ◽  
Author(s):  
David J. Lugo ◽  
Armando J. Blanco
Keyword(s):  
Fluid Dynamics ◽  
Computational Fluid Dynamics ◽  
Shear Stress ◽  
Power Law ◽  
Rotation Speed ◽  
Oil Industry ◽  
Annular Space ◽  
Bingham Plastic ◽  
Static Fluid ◽  
Pipe Rotation

Some industrials processes are associated with flow of non-Newtonian fluids in annular spaces created between parallel pipes. Examples are found in oil industry and food industrial processing. Depending on relative position of both axes, a concentric or eccentric annular space is created. In some typical applications the fluid rheology non-Newtonian and models such as Bingham Plastic or Power Law are required for adequate representation of internal deformations of fluid elements when shear stresses are applied. Depending on annulus eccentricity high resistance can be opposed to flow on narrowest section, including the possibility of having static or quasi-static fluid close to the internal annulus walls. In order to remove this static fluid, two different operations are usually proposed: pipe rotation and pipe reciprocation. In this way, less mobile fluid can be put in motion increasing shear stress. These operations are justified by experimental evidence exists. Scale experiments have been done and predictions for flow behavior in large facilities are extrapolated. However, in large facilities, as oil wells are highly pressurized and they are very deep, it is almost impossible to verify if the whole fluid is mobile and no by-pass fluid remains in the narrowest section of annular space. So, Computational Fluid Dynamics constitutes an ideal technique for analyzing this kind of problem. In this paper, though a Computational Fluid Dynamics study we aim to evaluate the efficiency of pipe rotation and pipe reciprocation in static or quasi-static fluids for Bingham Plastic or Power Law fluid. In order to consider realistic scenarios, oil industry typical conditions are considered for fluid density, rheological parameters, flow rates, casing and hole sizes, and annulus eccentricity. The influence of the variables eccentricity and rotation speed, and the use of reciprocation in shear stress at walls, were used as a measure to evaluate efficiency in static fluid removal. The flow regime was considered laminar. Numerical model capability to reproduce accurately flow patterns in these conditions was assured by comparing it with others analytical-numerical solutions for concentric systems. Results show that both operations are effective for helping in static fluid remotion. However, notable increment for efficiency is observed for eccentricities below 60%. In particular, pipe rotation is effective when rotation speed is greater than 20 RPM for eccentricity greater than 40%. Below this limit, pipe reciprocation is more effective than pipe rotation, independently of the rheological model used to represent the fluid.

scholarly journals Treatment of Shear Stress versus Shear Rate Data for Natural Fiber Reinforced Polymer Composites: A Discussion

2019 ◽  
Vol 11 (1) ◽  
pp. 89-100
Author(s):  
K. Begum ◽  
M. A. Islam
Keyword(s):  
Shear Stress ◽  
Shear Rate ◽  
Power Law ◽  
Flow Index ◽  
Rate Data ◽  
Power Law Model ◽  
Fiber Reinforced ◽  
Melt Viscosity ◽  
Fiber Loading ◽  
Stress Dependent

The rheological properties of melt jute fiber reinforced polypropylene (PP) composites were conducted at constant shear stress. The measured shear stress and shear rate data are fitted to a power law model for measuring stress-independent melt viscosity of the composites. The viscosity increased with the increase of fiber loading and decreased with the rise of temperature. The flow behavior index, n was found to decrease with the increase of fiber loading and increase with the rise of temperature. The shear stress and shear rate data collected from different specialized research journals have also been fitted to the power law model to measure the stress-independent melt viscosity and flow index as in all the previous literatures viscosity is treated as stress dependent parameter. It was found that the dependence of the viscosity and the flow index observed from previous literature data with fiber loading and temperature was quite consistent with the present study.

A Rational Friction Factor Correlation for Laminar Fully-Developed Pipe Flows of Shear-Thinning Non-Newtonian Fluids

2021 ◽  
Author(s):  
Robert Brewster
Keyword(s):  
Shear Rate ◽  
Friction Factor ◽  
Power Law ◽  
Fluid Model ◽  
Shear Thinning ◽  
Newtonian Fluids ◽  
Cross Model ◽  
Pipe Flows ◽  
Design Calculations ◽  
Friction Factor Correlation

Abstract A friction factor correlation for laminar, hydrodynamically fully-developed pipe flows of shear-thinning non-Newtonian fluids is derived through analysis and asymptotic considerations. The specific non-Newtonian fluid model used is the Extended Modified Power Law (EMPL) model, which is functionally equivalent to the Cross model. The EMPL model spans the entire shear rate range from the low to the high shear rate Newtonian regions, and includes the intermediate shear rate power law region. The friction factor correlation has an explicit form and is a function of three dimensionless parameters, making it well-suited to design calculations. The overall accuracy of the correlation is 6.6%, though it is much better in most cases. Graphical results for the correlation, and deviations with respect to high-accuracy numerical calculations are presented and discussed.

Axial Laminar Flow of Non-Newtonian Fluids in Narrow Eccentric Annuli

1965 ◽  
Vol 5 (04) ◽  
pp. 277-280 ◽  
Author(s):  
Robert D. Vaughn
Keyword(s):  
Laminar Flow ◽  
Power Law ◽  
Flow Rate ◽  
Pressure Loss ◽  
Large Diameter ◽  
Experimental Studies ◽  
Small Diameter ◽  
Journal Bearings ◽  
Newtonian Fluids ◽  
Eccentric Annuli

Abstract The analysis of laminar flow of power-law non- Newtonian fluids in narrow, eccentric annuli is employed in this paper to discuss the problems of lubricant flow in journal bearings and of errors introduced by eccentricity in experimental studies with concentric annuli on extruders and wellbore annuli. The velocity profile and pressure loss-flow rate equations are developed for the laminar flow region. In addition, the expected error in flow rate and pressure-loss measurements for concentric annuli as a result of eccentricity is determined. For example, a 10 per cent displacement of the core of an almost concentric annulus would cause a 1.8 per cent decrease in the observed pressure loss for a fluid with a power-law exponent n of 0.25. The corresponding increase in the observed volumetric flow rate would be 7.5 per cent. Introduction Non-Newtonianism and eccentricity occur simultaneously in two engineering problems:flow of lubricants in journal-bearings and pressure-reducing bushings, andflow of non-Newtonian fluids in plastic extruders and wellbore annuli. The lubricants used for moving parts are often non-Newtonian in character - often they are plastic in behavior. A solution to the problem of flow of non-Newtonian fluids in narrow eccentric annuli is particularly pertinent to this problem. In all experimental studies of laminar flow of fluids in concentric annuli, such as in extruders and well casings, the error due to eccentricity must be estimated or studied. A number of publications have dealt with this problem for Newtonian fluids; however, I am not aware of work for non-Newtonian fluids. This work is directed to the non-Newtonian problem. Before the solution to the problem is given, the pertinent conclusions from the work on Newtonian fluids will be reviewed. Heyda and Redberger and Charles have published general solutions to the problem of the laminar flow of Newtonian fluids in eccentric annuli, apparently without knowing of the earlier work of Caldwell and Bairstow and Berry, which is reported by Dryden, et al. Although several mathematical routes are encompassed by the work of these authors, the results appear to be equivalent. Redberger and Charles show that the error caused by eccentricity in concentric annuli is negligible for small diameter ratios (K less than 0.5); however, for large diameter ratios (K - 1), the error in the predicted flow rate can be as great as 100 per cent or more. Partial solutions to the problem are available from the work of Dryden, Tao and Donovan and Piercy, et al. Tao and Donovan examined the case of flow in narrow, eccentric annuli (K - 1) with and without rotation of the annular core. These authors also reviewed previous work on this subject and verified their approach with experimental data. Dryden gives the solution for the limiting case of complete eccentricity or tangency. Piercy, et al. published an early solution to the problem of narrow eccentric annular flow. The conclusions of Redberger and Charles and the experimental proof of Tao and Donovans both suggest that the region of large diameter ratios (K - 1) is of main interest and that the parallel planes approximation to the solution in this region is satisfactory. This method will now be extended to the laminar flow of non-Newtonian fluids in narrow eccentric annuli. THEORETICAL SOLUTION The geometrical aspects of the problem are illustrated in Fig. 1. To represent the non-Newtonian fluid the power-law model was selected. (1) This model has many disadvantages which have been pointed out; nevertheless, As simplicity, its frequent and wide applicability justify its use in this work. Fredrickson and Birds and Savins have used it as a basis for a theoretical study of laminar flow of non-Newtonian fluids in concentric annuli. SPEJ P. 277ˆ

Study on Design the Stop Pipe Diameter of Packaging Power-Law Fluid

2012 ◽  
Vol 198-199 ◽  
pp. 128-132
Author(s):  
Yong Ding ◽  
Fu Xin Yang ◽  
Jian Qiang Bao
Keyword(s):  
Shear Stress ◽  
Laminar Flow ◽  
Power Law ◽  
Flow Rate ◽  
Pipe Diameter ◽  
Power Law Fluid ◽  
Fluid Properties ◽  
Acceleration Of Gravity

The distribution of the speed and shear stress in power-law fluid with the laminar flow in the pipe were analyzed in this paper, then, the flow rate was calculated. Moreover, the stop pipe diameter was designed by calculating the balance of shear stress of power-law fluid in the pipe and the gravity of filling fluid. The conclusion: Ideal stop pipe diameter of power-law fluid is related to fluid properties, pressure and the acceleration of gravity.

A Theory of Transitional and Turbulent Flow of Non-Newtonian Slurries Between Flat Parallel Plates

1971 ◽  
Vol 11 (01) ◽  
pp. 52-56 ◽  
Author(s):  
Richard W. Hanks ◽  
Maheshkumar P. Valia
Keyword(s):  
Reynolds Number ◽  
Turbulent Flow ◽  
Critical Reynolds Number ◽  
Frictional Resistance ◽  
Newtonian Fluids ◽  
Transitional Flow ◽  
Parallel Plates ◽  
Bingham Plastic ◽  
Turbulent Transition ◽  
Large Width

Abstract A theoretical model is developed which Permits prediction of velocity profiles and frictional prediction of velocity profiles and frictional resistance factors for the isothermal flow of Bingham plastic non-Newtonian slurries in laminar, transitional, and turbulent flow between that parallel walls, in rectangular ducts of large width-to-height ratios, or in concentric annuli with radius ratios approaching unity. The theory is tested with available frictional resistance data for a range of Hedstrom numbers from 10(4) to 10(8) and a set of theoretical design curves of friction factor vs Reynolds number is developed. The model indices that for certain ranges of Hedstrom number (the non-Newtonian index) turbulence is suppressed relative to Newtonian flow behavior, whereas for other ranges of Hedstrom number, the converse is true. Introduction The handling of non-Newtonian fluids in turbulent motion is an important operation in many modern technological processes. Despite this fact, however, little has been done to develop models which are comparable to those available for Newtonian turbulent flow. In particular, a model of the transitional flow regime is notably lacking. Recently, a theory of laminar-turbulent transition for non-Newtonian slurries flowing in pipes and parallel plates was presented. A theory of parallel plates was presented. A theory of transitional and turbulent flow of Newtonian fluids in pipes and parallel plate ducts has also recently been developed. This theory permits the analytic calculation of the friction factor-Reynolds number curves as a continuous function of Reynolds number from the critical Reynolds number of laminar turbulent transition to any condition of turbulent flow. In this paper these two theories will be combined in order to develop a theory for the transitional and turbulent flow of non-Newtonian slurries in parallel plate ducts, rectangular ducts of large width-to-height ratio, or concentric annuli with radius ratios approaching unity. THEORETICAL ANALYSIS The rheological model which will be used to represent the non-Newtonian slurry behavior is the linear Bingham plastic model, ..............(1) ............(2) For this model the laminar flow curve is given by ..............(3) where q = 2v/b, b is one-half the distance between the plates, w = b(−dp/dz) is the wall shear stress, and D = o/ w. The end of the laminax flow, region is determined by the equations ........(4) .........(5) where N Rec = 4bp vc/ p is the critical Reynolds number, Dc is the critical transitional value of D and N He -16bp o/ p is the Hedstrom number expressed in terms of the hydraulic diameter for parallel plates. parallel plates. The calculation of the transitional flow field for this type of fluid will be based upon the model developed by Hanks for Newtonian fluids. SPEJ P. 52

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