1. VG Pienaar, PT Slatter, NJ Alderman+ & NI Heywood+
A Review of Frictional Pressure Losses for Flow of settling slurries and non-settling Non-Newtonian Fluids through Pipe Bends
VG Pienaar, PT Slatter, NJ Alderman+ & NI Heywood+
Cape Technikon, RSA
+ Hyprotech, UK

2. OVERVIEW OF PRESENTATION
Objectives of Review
Classification of Bends
Definition of Loss Coefficient
Laminar flow
Laminar/turbulent transition
Turbulent flow
Summary of work done
Conclusions

3. OBJECTIVES
To identify, collate, review, categorise information that has been published on frictional pressure losses arising from the flow of various types of non-Newtonian fluids and settling coarse particle slurries flowing through various pipe bends.
To compare experimental results with empirical predictive equations 

4. CLASSIFICATION OF BENDS
Circular-arc bend is a pipe that is smoothly curved over its bend angle
Elbow is defined as a bend where the diameter changes along the flow length
It was important to classify the information and bends and elbows were defined for the purpose of this document as:

5. DEFINITION OF LOSS COEFFICIENT
kfitt is the non-dimensionalised difference in overall pressure between the ends of two long straight pipes when there is no fitting and when the real fitting is installed.
pfitt is the pressure loss across the fitting. The flow lengths over which pressure losses occur starts from a few diameters upstream to several pipe diameters downstream of the actual length of the fitting. This is known as the region of influence or interference.
  is the density of the fluid. In the case of settling slurries it can be based on either the in-situ or delivered concentration and this also needs to be specified.
V is the mean flow velocity in the pipe. If there is a change in pipe diameter, the convention is to use the higher mean flow velocity of either the upstream or the downstream pipe.

6. EXPERIMENTAL PROCEDURE
The preferred experimental procedure
is to measure the overall friction loss
in a system made up of two pieces of straight pipe
connected in series by the fitting
and to subtract the losses in the straight pipes
from the total loss measured to obtain the loss due to the fitting.
Difficulty of distinguishing between the developed and incompletely developed flow
The preferred experimental procedure is to measure the overall friction loss in a system made up of two pieces of straight pipe connected in series by the fitting and to subtract the losses in the straight pipes from the total loss measured to obtain the loss due to the fitting.
This method is preferred over that of measuring the loss just across the fitting because of the difficulty of distinguishing between the developed and incompletely developed flow as the fluid enters and leaves the fitting.
Non-Newtonian fluids were modelled using power law, Bingham plastic or viscoplastic models.

7. LAMINAR FLOW TURBULENT FLOW
Loss coefficient, k is inversely proportional to the Reynolds number
TURBULENT FLOW
Loss coefficient k is dependent on the Reynolds number, but in comparison to laminar flow it can be taken as reasonably constant

8. LAMINAR/TURBULENT TRANSITION
Laminar/turbulent transition does not exist at a specific point, but is a region.
Experimental results for laminar/turbulent transition in bends are taken only from experimental results that were presented in the form of kbend vs Re.
Recrit is often taken as the intersection of the lines for laminar and turbulent flow. When determined this way, Recrit often lies half way between the upper and lower critical Re.
Equations to predict Recrit have been developed for Newtonian fluids flowing in long radius bends by Srinivasan and Ward-Smith.
Results from Ma and Kittredge and Rowley were used for comparison.
The values from the deviation method seem to be the lower limit whereas those determined by the intersection method are the upper limit and closer to calculated values.
Recrit is different for bends of different diameters
There is an increase in Recrit with increasing rc/D values for rc/D less or equal to 8.

9. SUMMARY OF WORK DONE Author Work Type of Bend rc/D Conclusions
Kittredge & Rowley,1957
Comparison of smooth bends with elbow
Smooth bend 
90 Elbow
2.26; 3.04; 6.53; 11.71 -
No consistent trend of k as a function of rc/D for smooth bend.
Little difference between 90 elbow and 90 smooth bends at Re2000.
Wildmann, Ekmann & Klinzing,1984
Smooth bend
9; 18
The 90 elbow produced less energy losses for 60% concentration and the smooth bend for the 65% concentration.
Steffe, Mohammed & Ford,1984
Flow through short radius bend
Short radius bend
2.1
Determined loss coefficient data for laminar and turbulent flow through a bends.

10. SUMMARY OF WORK DONE cont..
Author
Work
Type of Bend
rc/D
Conclusions
Das, Biswas & Mitra
1991
Comparison of smooth bends of various radius ratios
45 Bend
90 Bend
135 Bend
180 Bend
6.289
2.658
3.395
5.579
A generalised correlation has been developed for predicting the pressure drop across the bend of any angle in the horizontal plane for non-Newtonian pseudoplastic fluid in laminar flow.
Bojcic, Khan & Broadfoot
1997
Comparison of smooth bends of various radius ratios and angle
Long radius bend
Short radius bend
6.72
1.47
1.84
For flow below ReMR = 0.1 – 0.5 no pressure loss occurs due to the directional change of the fluid, and the loss is the same as for the same length of straight pipe as the fitting.
Turian et al.
1998
Comparison of smooth bends of various rc/D
45 Elbow
180 Elbow -
1.4; 4.3; 8.2; 12.5
Determined loss coefficient data for laminar and turbulent flow through various bends.

11. SUMMARY OF WORK DONE cont..
Author
Work
Type of Bend
rc/D
Conclusions
Edwards, Jadallah & Smith,1985
Comparison of bend of different size
90 Elbow
1 inch
2 inch -
The loss coefficient is independent of geometry for both laminar and turbulent flow.
Motley et al.
1985
1 inch,2inch
3 inch
Loss coefficient data for the different size elbows was found to be dependent on size and is in contrast with the above case.

12. COMPARISON OF FRICTION FACTORS FOR BENDS OF VARIOUS RADIUS RATIOS
The correlation of Das et al was used to calculate the frictional losses for bends of various radius ratios and compared with experimental results of Bojcic et al. Good agreement was found. The length of the bend is required to calculate the friction factor for the bend and this was not given by other researchers who presented their work in the form of kbend vs Re.
Showing that indeed below Re =0.1, the losses due to change in direction is negligible and only due to that of the same length of straight pipe.

13. COMPARISON OF kbend FOR 90 DEGREE BENDS
Turian et al and Steffe determined values of Cbend for bends of various radius ratios were plotted. Superimposed on this plot are the experimental work of Das et al and Kittredge and Rowley and Steffe et al.
There is not enough information to conclude on the validity of the results or correlations

14. COMPARISON OF kbend FOR 90 DEGREE ELBOWS
The results of Edwards et al and Motley et al were in direct contradiction. Edwards found that the loss coefficient data was the same for different size elbows in both laminar and turbulent flow, whereas Motley et al found that the losses was dependent on the size of the elbow.

15. GUIDELINES FOR CHOOSING BENDS
Advantageous to use a bend with as large a radius of curvature as possible because the pressure drop becomes a maximum at small values of radius ratio and reduces as the radius of curvature increases (19).
Effect of rc for the total pressure drop of the pipeline involving a pipe bend is very small (20)
The effect of particles for the pressure drop in vertical pipe bends is larger than that for horizontal bends because the effect of gravitational force becomes very large when the particles enter the vertical section from the horizontal section (20)
For turbulent flow of Newtonian fluids for a 45 bend, the total friction loss is about 65% of the loss for a 90 bend of a proportional number of segments and for a 180 bend the loss is approximately 140% of that for a 90 bend (12)
According to Ayukawa (19) it is advantageous to use a bend with as large a radius of curvature as possible because the pressure drop becomes a maximum at small values of radius ratio and reduces as the radius of curvature increases.
According to Toda et al. (20) the effect of rc for the total pressure drop of the pipeline involving a pipe bend is very small, and the optimum radius of curvature will be decided by other factors in most cases.
The effect of particles for the pressure drop in vertical pipe bends is larger than that for horizontal bends because the effect of gravitational force becomes very large when the particles enter the vertical section from the horizontal section (Toda et al.)
For turbulent flow of Newtonian fluids for a 45 bend, the total friction loss is about 65% of the loss for a 90 bend of a proportional number of segments and for a 180 bend the loss is approximately 140% of that for a 90 bend (Perry, 12).

16. NEWTONIAN FLUIDS LAMINAR TURBULENT
The loss coefficient becomes inversely proportional to the Reynolds number.
Values of Cbend from Hooper 1981.
kbend =Cbend/Re
The loss coefficient is independent of the Reynolds number for the purposes of this review.
It is higher for smaller fittings and the effect of surface roughness becomes important.
List of kbend values from Hooper 1981, ESDU (1983), Miller (1978) and ASHRAE (1989).
The loss coefficient becomes inversely proportional to the Reynolds number at low Reynolds numbers.
Values of Cbend from Hooper 1981.
kbend =Cbend/Re
The loss coefficient is independent of the Reynolds number for the purposes of this review.
It is higher for smaller fittings and the effect of surface roughness becomes important. List of kbend values from Hooper 1981, ESDU (1983), Miller (1978) and ASHRAE (1989).

17. NON-NEWTONIAN SLURRIES
LAMINAR
TURBULENT
Cbend is similar to Newtonian fluids.
Non-Newtonian behaviour accounted for by Re
Various bends tested by Turian et al. (1998) and Edwards (1985),(Bojcic, 1997).
ReMR = 0.1 – 0.5 no pressure loss occurs due to the directional change of the fluid
Above this ReMR, an additional pressure loss exists due to directional change
Equations to calculate the frictional losses suggested by Wildman et al. (1984), Das et al. (1991) and Bojcic et al. (1997).
Most experimental work suggests that the loss coefficient is the same as Newtonian fluids.
Cbend is similar to that for Newtonian fluids.
Non-Newtonian behaviour can be accounted for by using the Reynolds number best
describing the rheology of the fluid.
Various bends tested by Turian et al. (1998) and Edwards (1985).
For flow below ReMR = 0.1 – 0.5 (depending on fitting) no pressure loss occurs due to the directional change of the fluid, and the loss is the same as for the same length of straight pipe as the fitting. Above this ReMR, an additional pressure loss exists due to directional change (Bojcic, 1997).
Equations to calculate the pressure drop were suggested by Wildman et al. (1984), Das et al. (1991) and Bojcic et al. (1997).

18. SETTLING SLURRIES LAMINAR TURBULENT Not transported in laminar flow.
kbend vs (rc/D) for flow of slurries through 90 bends attains a minimum.
Particle size effects a discernible influence on friction losses.
According to Mukhtar (1995), losses for multisized particles are less than that for water. Drag reduction at certain particle sizes.
kbend vs (rc/D) for flow of slurries through 90 bends attains a minimum as in the case of single phase fluids.
Particle size effects for fine non-colloidal particles do not seem to have a discernible influence on friction losses.
According to Mukhtar (1995), losses for multisized particles are less than that for water. Drag reduction at certain particle sizes.

19. CONCLUSIONS
Frictional pressure loss data has been reviewed for both bends and elbows
Data for both turbulent and laminar flow of non-Newtonian slurries have been presented in a single document
The breakdown of laminar flow in bends has been discussed
The data has been compared with empirical predictive equations
Comparisons have identified the many gaps in loss coefficient information which still require filling