Two-Phase (Solid-Liquid) Flow in Inclined Pipes

Two-Phase (Solid-Liquid) Flow in Inclined Pipes

Evaldo Miranda Coiado

Victor Emanuel M. G. Diniz

Universidade Estadual de Campinas

Faculdade de Engenharia Civil

Cidade Universitária "Zeferino Vaz"

Distrito de Barão Geraldo

Caixa Postal 6021

13083-970 Campinas, SP. Brazil

[e-mail], [e-mail]

This paper presents an experimental research about the behavior of two-phase flows in inclined pipes. The inclination angle varied from 5° to 45° and the slurry solid concentration varied up to 15%. It was concluded that the head losses of the downward sloping pipe flow are always lower than the head losses of the horizontal flow and these are always lower than the head losses of the upward sloping pipe flow, regardless the concentration and inclination angle. It was possible to develop empirical equations to calculate the head losses of the horizontal flow and the upward and downward sloping pipe flows.

Keywords: Head loss, settling mixtures, inclined pipes

Introduction

Most of the applications of hydraulic transport in the past have been in the minerals industries. Frequently, such industrial activities are located in remote locations with insufficient road or rail infrastructure, and the use of pipeline transport has been the preferred and most economic method of transporting large quantities of minerals over long distances and through difficult terrain. Almost invariably the solid particles are denser than the conveying liquid. Therefore in horizontal pipelines the solid particles travel in the lower part of the cross-section, with liquid above it. This behavior becomes a special subject of study.

The design of a long-distance installation requires the knowledge of the two-phase (solid-liquid) flow behavior in horizontal pipe and in inclined pipe.

Most published material about two-phase (solid-liquid) flow are related to horizontal pipe flow. There are a limited number of studies concerning the effect of pipe inclination on two-phase flow energy loss. Usually, for the design of these transports lines, the head loss evaluation of the inclined pipeline portion is based upon a composition between the horizontal head losses and the vertical head losses. A trigonometric decomposition of the head losses concerning the vertical length to the desired plan can also be made. Usually, the designers do not take into account the pipe inclination. They consider the whole transport line as horizontal.

The main objective of the present study is to study the flow head losses in upward and downward sloping lengths. The influence of flow velocity, slurry concentration and the sloping pipe inclination angle in the head loss was investigated. The head losses determined for the horizontal portion of the pipeline were compared with available correlations found in the literature

A pipe loop system was built. This loop system allows an easy variation of the flow velocity, the slurry concentration and the pipe inclination.

Nomenclature

C = slurry concentration in volume (%)

dr = relative density of solid

d_L = density of liquid phase (kg/m³)

e = ( å|100.(J_author – J_eq.) / J_eq. | )/ N = relative error (%)

J = head loss of the liquid phase (m/m)

Ja = head loss of the slurry upward sloping flow (m/m)

Jd = head loss of the slurry downward sloping flow (m/m)

Jh = head loss of the slurry horizontal flow (m/m)

Jm = head loss of the slurry (m/m)

Js = head loss of the solid phase (m/m)

N = number of measurements

V = flow average velocity (m/s)

a = pipe inclination angle (rad)

s = standard deviation.

Bibliographic References

The flow of solid-liquid slurries in pipes differs from the flow of homogeneous liquids in a number of ways. With liquids the complete range of velocities is possible, and the nature of the flow (i.e., laminar, transition, or turbulent) can be characterized from a knowledge of the physical properties of the fluid and the pipe system. Characterization of slurry flow is not as simple as for liquid flow for two reasons. Firstly, there are, superimposed on the properties of the liquid, the properties of the solid particles to be accounted for and also the effect of the particles on the mixture properties. Secondly, depending on the particular conditions, a range of slurry behavior is possible, this latter point being best illustrated by consideration of the two extremes of slurry flow which can be identified, Wasp et al., 1999.

Homogeneous flow is the term given to system in which the solids are uniformly distributed throughout the liquid medium. Homogeneous flow, or a close approximation to it, is encountered in slurries of high solids concentrations and fine particle sizes. The presence of the solids can have a significant effect on the system properties, usually resulting in a sharp increase in viscosity as compared to that of the carrier fluid. In heterogeneous flow systems, solids are not evenly distributed and in horizontal flow, pronounced concentration gradient exist along the vertical axis of the pipe, even at high velocities. Particle inertial effects are significant, i.e., the fluid and solid phases to a large extent retain their separate identities, and the increase in the system viscosity over that of the carrier liquid is usually quite small. Heterogeneous slurries tend to be of lower solids concentrations and have larger particles sizes than homogeneous slurries, Richardson, et al., 1999.

The heterogeneous suspension regime is the most important mode of transport of granular materials by pipelines, because the maximum amount of solids is transported per unit energy input. There is in the literature a considerable number of publications about heterogeneous suspension regime in horizontal pipe. On the other hand, the number of articles concerning heterogeneous regime in non-horizontal pipeline is reduced, and there is a lack of information about the use of the equations which permit to compute the head loss.

In vertical pipes the velocity of solids for upward flow is less than the fluid velocity and it is greater for downward flow. The difference is approximately the value of the settling velocity, approximately, Raudkivi, 1989.

The head loss of the solid-liquid slurries in inclined pipes according to Worster and Denny, 1955 is:

Einstein and Graf, 1966, considering flow in riser, suggested to calculate the head loss in vertical pipes by the following relation:

Graf, 1972, has modified Eq. (2) to apply for inclined conduits, by

According to Durand & Gibert, 1960, in Kao and Hwang, 1979, the effect of solid presence in the head loss gradient in an inclined pipe in relation to a horizontal pipe is given by:

It was not found, in the literature, experimental investigation validating equations (1), (2), (3), and (4), so further discussion about they, it is not relevant at the present moment.

Some researchers studied two-phase (solid-liquid) flow in inclined pipes, but they did not developed equations. They are: Wilson and Tse, 1984, Brigham et al., in Streeter, 1961, Kao and Hwang, 1979, and Nakae, 1990.

Materials and Methods

To reach the objectives proposed in this research, a pipe loop system, as shown in Figure 1, was constructed in the laboratory of the Department of Hydric Resources, Civil Engineering, University of Campinas.

The head loss measurements for the sand-water slurry flow were carried out in pipelines of D=75 mm. It was used an uniform sand with d₅₀=0.20mm; dr=2.68, the concentrations were up 15 percent in volume. The pipe inclination angle used was varied from 5.5^o to 45^o .

Head loss was measured in a horizontal pipeline allowing to verify the effect of pipe inclination on slurry transport and energy loss.

The slurries (mixture of sand-water) were prepared in the main reservoir, showed in Figure 1. The mixture was maintained homogeneous in the main reservoir by the use of an auxiliary pump. Then, the homogeneous mixture was pumped, through the pipeline, by the main pump. The pipeline was made up of horizontal and inclined pipes.

During the slurry flow through the pipeline, for different velocities and concentrations of the mixture, the following parameters were measured: a) the head losses in the horizontal, upward, and downward sloping lengths; b) the discharge; c) the concentration of the mixture.

The discharge was measured by a magnetic flow transmitter (meter flow). The concentration of the mixture was determined using a reservoir to measure the volume and one balance. To this purpose, the mixture flow was deviated to the mensuration reservoir.

The head losses in the horizontal, upward, and downward sloping lengths were measured by different pressure transmitters.

Mathematical Description

Three empirical equations were developed to calculate the head losses of the water-sand slurry in function of the flow average velocity, the sand concentration and the inclination angle of upward and downward flows in inclined pipes.

The equations were obtained by several curve fittings. To illustrate the procedure, the flow through a horizontal pipeline and the upward sloping part of a pipeline will be used. The same procedure was used with the downward sloping pipe.

Upward Sloping Length Flow

It is presented the fitted functions to points Jm x V for the water-sand slurry flows with concentrations of 5%, 10% and 15% in the upward and downward sloping lengths for inclination angles of 5.5° , 11° , 22.5° , 34° and 45° .

Fit 1: Jm x V for five inclination angles a1, a2, a3, a4, and a5 and three concentrations with values C1, C2 and C3.

Jaij, where i= angle and j= concentration

For concentration with value C1:

For concentration with value C2:

For concentration with value C3:

The coefficients of the first set of curves were obtained. That is, by keeping the concentration constant, for a given concentration, for example 5%, fitted functions were obtained for each coefficient versus the inclination angle. This procedure was repeated for the concentrations of 10% and 15%.

Fit 2: the coefficients aij, bij and cij were fitted as function of the angles a1, a2, a3, a4 and a5 for each concentration.

For concentration with value C1:

For concentration with value C2:

For concentration with value C3:

Results from fittings 1 and 2:

For concentration with value C1:

For concentration with value C2:

For concentration with value C3:

Coefficients of the second set of curves were obtained and fitted functions were determined for these coefficients versus the concentrations.

Fit 3: the coefficients dij, eij and fij were fitted in function of the concentrations.

Results from fittings 1, 2 and 3:

Horizontal Length Flow

Next, the fitted functions to points Jm x V for the water-sand slurry flows with concentrations of 5%, 10% and 15% in the horizontal length is presented.

Fit 1: Jm x V for three concentrations with values C1, C2 and C3.

Jj, where j = concentration

For concentration with value C1:

For concentration with value C2:

For concentration with value C3:

The coefficients of the first fittings were determined and fitted functions were obtained for these coefficients versus the concentrations.

Fit 2: the coefficients aj, bj and cj were fitted in function of the concentrations.

Results from fittings 1 and 2:

Results and Analysis

After all fittings were done, the following three equations were developed to calculate the head losses of the water-sand slurry for concentrations from 5% to 15%, inclinations from 5.5° to 45° and flow average velocity from 2.0 m/s to 5.5 m/s:

a) Horizontal length flow

b) Upward sloping length flow

c) Downward sloping length flow

To validate the developed equations, they were compared to the experimental collected data. Figures from 2 to10 illustrate the results for the upward and downward sloping flow and horizontal flow for water-sand slurries with concentrations of 5%, 10% and 15%. A curve family was obtained for each graph varying the inclination angle. Each curve refers to an angle. It was observed that in a general way, the developed equations fit well to the experimental collected data.

From Graf's equation (1972), Worster and Denny's equation (1955) and the equations developed in this research, two charts, (Charts 1, and 2), were elaborated with the relative errors between the head losses provided by the authors' equations and the head losses provided by the equations developed.

For water-sand slurry flow in the horizontal, upward and downward lengths, it was observed that the authors' equations provide values of head losses which deviate considerably to the ones obtained by the present work. One can observe that relative errors increase if angle and/or concentration increase.

For the downward sloping length flow, the values of the head losses obtained by the authors' equations are very distant from those obtained by the present work. This can be explained by the fact that the authors' equations provide the same values of head losses for the upward and downward lengths, and this was not observed during the experimental tests. It was observed in Figures 2 to 4 that values of the head losses increase as the concentration and the inclination angle increase for the upward length flow. On the other hand, values of the head losses decrease as the concentration and the inclination angle increase for the downward length flow, (Figures 5 to 7).

Charts 1, and 2 present relative errors between the authors' equation and the equations developed at the present work.

Chart 3 presents relative errors between experimental data and computed values using the equations developed at present work. It can be seen that the developed equations give reasonably results. The relative errors are smaller than two percent.

Measurement Uncertainties and Errors

Care was taken at the present work, so that low measurement errors of the parameters were obtained.

The discharge was measured by a magnetic flow transmitters (meter flow). System accuracy is ± 0.5%. Repeatability: ± 0.1% of reading. Response time: 0.2 seconds maximum response to step change in input. Stability: ± 0.1% of rate over six months. Ambient temperature effect: ± 1% per 37 ⁰C.

The head losses were measured by differential pressure transmitter, with accuracy ± 0.25% for calibrated span. Combined effects of linearity, hysteresis, and repeatability were included. Temperature effect is less than ± 1.5% of upper range limit per 55 ⁰C. Static pressure effect is less than ± 0.5%.

Each measurement of discharge and head loss was read, approximately, 138 times. Chart 4 and Chart 5 illustrate an example of measurement of head loss. These charts present the head losses and their respective standard deviations (s) and confidence intervals.

Chart 5

It can be seen, in Chart 4, that the values of standard deviation increase if the discharge values and concentration values also increase. Therefore, larger values of (s) correspond to cases in which the "scatter" in the measurement values is larger, resulting in larger precision error.

Conclusions

Based on the collected experimental data, the adopted methods and the experimental conditions in which the tests were made, the following conclusions can be made:

The head losses values for the downward sloping pipes are always lower than the head losses for the horizontal pipe, and these are always lower than the head losses for the upward sloping pipes, regardless the inclination angles and concentrations.

For the downward sloping length water-sand slurry flow, the presence of sand decreases the head losses values as the inclination angle and concentration increase. Negative pressure values were obtained for the three tested concentrations, with inclination angles of 22.5° , 34° and 45° .

For the water-sand slurry flow in the horizontal pipe, the presence of sand increases the values of the head losses as the concentration increases.

For the upward sloping length water-sand slurry flow, the presence of sand increases the values of the head losses as the inclination angle and concentration increase.

For the horizontal, upward, and downward water-sand slurry flows, it was observed that Graf's equation (1972) and Worster and Denny's equation (1955) provide values of head losses which deviate considerably to the ones obtained by the equations developed in the present work.

The developed equations give reasonably results. The relative errors are smaller than two percent.

Article received: September, 1998. Technical Editor: Atila P. Silva Freire

• Einstein, H. A. and Graf, W. H., 1966. Lopp System for Measuring Sand-Water Mixture. Proc. Am. Soc. Civil Engrs., vol. 92, HY1.
• Graf, W. H.,1972, "Hidraulic of Sediment Transport", McGraw-Hill Book Company, New York, USA, 513p.
• Kao, D. T. Y. and Hwang, L.Y.,1979, "Critical Slope for Slurry Pipeline Transporting Coal and Other Solid Particles", Proceedings of the 6th International Conference on the Hydraulic Transport of Solids in Pipes, BHRA Fluid Engineering, Canterbury, U. K., pp. 57-74.
• Nakae, T.,1990, "Loop System to Study the Flow of Mixtures (Sand and Water) in Pipes", (In Portuguese), Master Thesis, State University of São Paulo, São Paulo, S.P., Brazil, 152p.
• Raudkivi, A. J., 1989 – Loose Boundary Hydraulics. Pergamon Press, New York.
• Richardson, J. F., Chhabra, R. P., Khan, A. R., 1999 – Multiphase flow ou non-Nextonian fluids in horizontal pipes. Slurry Handling and Pipeline Transport. Hidrotransport 14. Maastrich. Netherlands.
• Streeter,V.L. 1961, "Handbook of Fluid Dynamics", McGraw–Hill BooK Company, New York, U.S.A, 1215p.
• Wasp, E. J.; Kenny, J. P.; Gandhi, R. L. 1999. "Solid-Liquid Flow Slurry Pipeline Transportation". Series on Bulk Materials Handling. International Standard Book Number. Trans Tech Publications. Germany.
• Wilson, K. C. and Tse, J. K. P., 1984. Deposition Limit for Coarse-Particle Transport in Inclined Pipes. 9th International Conference on Hydraulic Transport of Solids in Pipes. Rome, Italy: 17-19.
• Worster, R. C. and Denny, D. F.,1955, " The Hydraulic Transport of Solid Material in Pipes", Proceedings of. Institute of Mechanical Enginnering. Vol. 169/32.

Publication Dates

• Publication in this collection
  19 Aug 2002
• Date of issue
  2001

History

• Received
  Sept 1998