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Figure 1. Cut view of a steam surface condenser

Experimental Data

Head loss for fully developed flow through a horizontal pipe of constant area is:

( )DeRDfLUh el /,2/2

=

Where,

L = pipe length

U = mean velocity

D = pipe inside diameter

eR = Reynolds number

e = pipe roughness.

( )DeRf e /, = unknown function.

The unknown function, ( )DeRf e /, is defined as thefriction factor, f,

( )DeRf e /,

therefore,

)1(2/2 KKKDfLUhl =

For laminar flow,fis found out to be independent of relative roughness (e/D),f= 64/ Re; for turbulent flowthe frictional factor is determined experimentally.

The results published by L.F.Moody are shown in Figure 2 (See Figure: Friction factor for fully developed

flow in circular pipes by L.F. Moody).

Nikuradse has obtained data for artificially roughened pipes. The observations were taken upto a Reynolds

number of 1x106. (Relative roughness e/D = 0.5/Relative smoothness).

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Figure 2. Friction factor for fully developed flow in circular pipes by L.F. Moody

Although the sand coated pipes used by Nikuradse differ appreciably from the commercial pipes available

in market, his data are extremely valuable since they provide a reliable basis for quantitative measurementof roughness effects. This has been achieved by introducing the concept of equivalent sand grain

roughness. The entire available data was first plotted by Stanton with logarithmic co-ordinates as shown in

Figure 3.

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Figure 4. Geometric model of pipe

A 2-D element FLUID141 has been used. Because of symmetry about the axis of the pipe, an axi-

symmetric model has been considered. Element length along flow direction (X-axis) has been kept as 0.010

m and 12 divisions have been made across the pipe radius with biasing towards the pipe wall. The modelcontains 13013 nodes and 12000 elements.

Figure 5 shows a line diagram of inlet part of pipe where the lines have been sized for generating theappropriate mesh. The top line represents wall and the bottom line represents the centerline of the pipe.

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Figure 5. Inlet end of pipe

The working fluid is water. Following constant properties have been considered:

Density = 997 Kg/m3

Absolute viscosity = 9 x 10-4N-s/m2

No flow conditions have been imposed along the wall. A constant velocity is given at inlet and zeropressure has been given at outlet.

Figure 6 indicates inlet end of meshed pipe. It shows inlet and wall boundary conditions. At center line thenormal component of velocity is zero.

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Intel Pentium 4 CPU 1.50GHz, 128 MB RAM, with Microsoft Windows 2000 was used for execution of

ANSYS FLOTRAN 7.0. For Reynolds numbers from 2000 to 1x108 (45 different values of Reynolds

number), with relative roughness (e/D) = 0.0004, it took continuous running of 21 hours 36 minutes.

During this time a total of 13481 global iterations were performed.

Analysis Results & Discussion

It has been confirmed that the pipe length of 10 meters is quite sufficient for flow to get fully developed for

laminar as well as turbulent flows for the range of Reynolds numbers considered.

Figure 7 and Figure 8 show the velocity profile of laminar flow, along the pipe length and normal to pipelength respectively, at a Reynolds number of 2000 and at a distance of 9 meters from inlet. Maximum

velocity along the length of the pipe is 0.1205 m/sec and maximum velocity normal to the length of the

pipe is 8.5 X 10-9 m/sec which is almost zero ensuring that the flow is fully developed.

Figure 7. Velocity profile along the pipe length at Reynolds number 2000 and at 9 metersfrom inlet

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Figure 8. Velocity profile normal to the pipe length at Reynolds number 2000 and at 9meters from inlet

Figure 9 shows the velocity vector plot at the outlet. This is fully developed laminar flow at a Reynolds

number of 2000.

Figure 9. Fully developed velocity vector plot at outlet at Reynolds number 2000

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Figure 10 and Figure 11 show the velocity profile of turbulent flow, with relative roughness e/D = 0.0 (i.e.

a smooth surface), along the pipe length and normal to pipe length respectively at Reynolds number 1x107

and at a distance of 9 meters from the inlet. Maximum velocity along the length of the pipe is 341.096

m/sec and maximum velocity normal to the length of pipe is 3.6 X 10-9 m/sec which is almost zero

ensuring that the flow is fully developed.

Figure 10. Velocity Profile along the pipe length at Reynolds number 1x107

and at 9 metersfrom inlet

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Figure 11. Velocity Profile normal to the pipe length at Reynolds number 1x107

and at 9meters from inlet

Figure 12 shows the velocity vector plot at the outlet. This is fully developed turbulent flow at Reynolds

number of 1x107.

Figure 12. Fully developed velocity vector plot at outlet at Reynolds number 1x107

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Similar exercises were done for other two relative roughness (e/D) values available in Moodys Graph,

Figure 2 and also for three relative roughness (e/D) values available in Nikuradses data as plotted by

Stanton, Figure 3.

Comparison with Moodys

Figure 13 shows the friction factor values as read from Moodys graph. For e/D=0.0 (Col 2), Reynoldsnumber is from 4000 to 1x107. For e/D = 0.00005 (Col 5), Reynolds number is from 9x104 to 1x108. For

e/D = 0.0004 (Col 8), Reynolds number is from 6000 to 1x108. These graphs have been shown in log-log

scale.

Figure 13. Friction factor values as read from Moodys graph

Figure 14 shows the friction factor values as per ANSYS FLOTRAN for corresponding values of e/D as

read from Moodys graph. ANSYS results have been shown for Reynolds numbers from 4000 to 1x108. ForReynolds numbers of 4000 to 1x104, a common straight line represents the value of friction factor

irrespective of the value of e/D.

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Figure 14. Friction factor values as per ANSYS FLOTRAN

Figure 15 shows the ANSYS values considering Reynolds numbers from 500 to 1x105 as laminar flow and

Reynolds numbers from 2000 to 1x108 as turbulent flow. This has been shown to see how ANSYS

FLOTRAN behaves for the extended range of Reynolds numbers.

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Figure 15. Friction factor values as per ANSYS FLOTRAN for extended Reynolds numberran

The plot between friction factor and Reynolds number for e/D = 0.0 (smooth pipe) is shown in Figure 16 in

log-log scale. In this figure, blue represents the experimental values obtained by L.F. Moodyand purple

represents values obtained from ANSYS.

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Figure 16. Friction factor Vs. Reynolds number plot for laminar & turbulent flow with e/D =0.0

In the laminar zone, for Reynolds numbers 700 to 2300, the results closely match the experimental values.

In the turbulent zone, large deviation is observed in Reynolds numbers from 4000 to 9000. This might be

contributed to the fact that ANSYS solver is applicable for Reynolds numbers of 104 and above. For

Reynolds numbers of 1x104

to 1x107

, for which Moodys values are available, the ANSYS results arealmost matching with the experimental values obtained by Moody. If the Moodys values are extrapolated

from 1x107

to 1x108, then the ANSYS results also closely match with these values.

The percentage deviation from the experimental data is shown in Figure 17. The x-axis is in log scale and

y-axis is in linear scale.

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Figure 17. % Deviation Vs. Reynolds number plot for laminar & turbulent flow with e/D =0.0

In laminar flow the maximum deviation is about 0.11%.

In turbulent flow, for Reynolds numbers 4000 to 9000, the deviation is large. It is about 62% at Reynoldsnumber 4000 and it has come down to 8.5% at Reynolds number 9000. For Reynolds numbers 1x104 to

1x107 the deviation is within about 4.4%. The maximum % deviation of 4.4% is at Reynolds number 3x106.The minimum deviation is 0.83% at Reynolds number 8x104. There is no negative deviation.

Similar to Figure 16 and Figure 17, curves for turbulent flow with different e/D values have been shown

from Figure 18 to Figure 21.

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Figure 18. Friction factor Vs. Reynolds number plot for turbulent flow with e/D = 0.00005

For e/D = 0.00005, Moodys values are available from Reynolds numbers 9x104 to 1x108. As shown in

Figure 18 & Figure 19, comparison has been made for this range only. From Figure 19, maximum negative

deviation is 8.7% at Reynolds number 2x106, zero deviation is at Reynolds number 9x104 & 9x106 and

maximum positive deviation is 13.1% at Reynolds number 5x107.

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Figure 19. % Deviation Vs. Reynolds number plot for turbulent flow with e/D = 0.00005

For e/D = 0.0004, Moodys values are available from Reynolds numbers 6x103 to 1x108. As shown in

Figure 20 & Figure 21, comparison has been made for this range only. From Figure 21, upto Reynolds

number of 9000, maximum positive deviation is 31.9% at Reynolds number of 6 x103. For Reynolds

number of 1x104

and onwards, maximum negative deviation is 10.4% at Reynolds number 2x105, zero

deviation is at Reynolds number 1x106 and maximum positive deviation is 13.4% at Reynolds number

1x108.

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Figure 20. Friction factor Vs. Reynolds number plot for turbulent flow with e/D = 0.000

Figure 21. % Deviation Vs. Reynolds number plot for turbulent flow with e/D = 0.0004

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Comparison with Nikuradse

Figure 22 shows the friction factor values as read from Nikuradse graph for Reynolds numbers from 4x103

to 1x106. For e/D = 0.0 (Col 2), for e/D = 0.000986 (Col 5) and for e/D = 0.001984 (Col 8). These graphs

have been shown in log-log scale.

Figure 22. Friction factor values as read from Nikuradse graph

Figure 23 shows the friction factor values as per ANSYS FLOTRAN for corresponding values of e/D as

read from Nikuradse graph. ANSYS results have been shown for Reynolds numbers from 4x103 to 1x106.

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Figure 23. Friction factor values as per ANSYS FLOTRAN

The plot between friction factor and Reynolds number for e/D = 0.0 (smooth pipe) is shown in Figure 24 in

log-log scale. In this figure blue represents the experimental values obtained by Nikuradse andpurple

represents values obtained from ANSYS.

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Figure 24. Friction factor Vs. Reynolds number plot for laminar & turbulent flow with e/D =0.0

In laminar zone, for the Reynolds numbers 700 to 2300, the results closely match the experimental values.

As observed earlier in turbulent zone, large deviation is observed in Reynolds numbers from 4000 to 9000.

For Reynolds numbers of 1x104 to 1x106, for which Nikuradse values are available, the ANSYS results

almost match with the experimental values. If the Nikuradse values are extrapolated from 1x106

to 1x108

,then also the ANSYS results closely match with these values.

The percentage deviation from the experimental data is shown in Figure 25. The x-axis is in log scale and

y-axis is in linear scale.

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Figure 25. % Deviation Vs. Reynolds number plot for laminar & turbulent flow with e/D =0.0

In laminar flow, the maximum deviation is about 0.11%.

In turbulent flow, for Reynolds numbers 4000 to 9000, the deviation is large. It is about 66.0% at Reynolds

number 4000 and it has come down to 6.2% at Reynolds number 9000. For Reynolds numbers 1x104 to

1x106

the deviation is within about 4.4% and this maximum deviation is at Reynolds number 3x105

. Theminimum deviation is -0.23% at Reynolds number 8x104. Maximum negative deviation is 2.4% at

Reynolds number 5x104.

Similar to Figure 24 and Figure 25, curves for turbulent flow with different e/D values have been shown

from Figure 26 to Figure 29.

For e/D = 0.000986, comparison has been shown in Figure 26 & Figure 27. From Figure 27, for Reynolds

numbers 4000 to 9000, the deviation is large. It is about 66.0% at Reynolds number 4000 and it has come

down to 6.2% at Reynolds number 9000. For Reynolds numbers 1x104 to 1x106 maximum negativedeviations is 2.2% at Reynolds number 5x104, minimum positive deviation is 0.35% at Reynolds number

1x104 and maximum positive deviation is 9.8 % at Reynolds number 1x106.

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Figure 26. Friction factor Vs. Reynolds number plot for turbulent flow with e/D = 0.000986(R/k = 507)

Figure 27. % Deviation Vs. Reynolds number plot for turbulent flow with e/D = 0.000986(R/k = 507)

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For e/D = 0.001984, comparison has been shown in Figure 28 & Figure 29. From Figure 29, for Reynolds

numbers 4000 to 9000, the deviation is large. It is about 66.0% at Reynolds number 4000 and it has come

down to 6.2% at Reynolds number 9000. For Reynolds numbers 1x104 to 1x106, maximum negative

deviation is 1.9% at Reynolds number 2x104, minimum positive deviation is 0.36% at Reynolds number

1x104 and maximum positive deviation is 13.2% at Reynolds number 8x105.

Figure 28. Friction factor Vs. Reynolds number plot for turbulent flow with e/D = 0.001984

(R/k = 252)

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Figure 29. % Deviation Vs. Reynolds number plot for turbulent flow with e/D = 0.001984(R/k = 252)

Conclusion

For laminar flow the deviation is within 0.11% and correlates well with the experimental values.

For turbulent flow, ANSYS FLOTRAN values could be considered from Reynolds number 1x104 and

onwards.

Comparison with Moodys

With relative roughness (e/D) = 0.0, for Reynolds numbers from 1x104 to 1x107, the deviation is within

about 4.4%. Also from Reynolds numbers 1x107 to 1x108 ANSYS FLOTRAN values are closely matching

the extrapolated values from Moodys curves. This shows very good agreement with experimental values.

For rough surfaces with e/D = 0.00005, for Reynolds numbers from 9x104 to 1x108 the deviation lies

between 8.7% to +13.1%. With e/D = 0.0004, for Reynolds numbers from 1x104 to 1x108 the deviation

lies between 10.4% to +13.4 %. This can be considered in good agreement with experimental values.

Although the shape of the friction factor curves obtained from FLOTRAN are not similar to that ofMoody's curves but the variation is small. This variation is quite reasonable for considering FLOTRAN

results for engineering purposes.

Comparison with Nikuradse

With relative roughness (e/D) = 0.0, for Reynolds numbers from 1x104 to 1x106, the deviation is withinabout 4.4%. Also from Reynolds numbers 1x106 to 1x108 ANSYS FLOTRAN values are closely matching

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the extrapolated values from Nikuradse Graph. This shows a very good agreement with experimental

values.

For rough surfaces the Reynolds number range is from 1x104 to 1x106. With e/D = 0.000986, the deviation

lies between 2.2% to +9.8%. With e/D = 0.001984, the deviation lies between 1.9% to +13.2 %. This can

be considered in good agreement with experimental values.

The shape of friction factor curves obtained by FLOTRAN are similar to that of Nikuradse's curves. The

variation is quite reasonable for considering FLOTRAN results for engineering purposes.

References

1. ANSYS Users Manual, ANSYS Inc., Southpointe, 275 Technology Drive, Canonsburg, PA15317, USA.

2. Heat Exchanger Institute, Inc. Standard for Steam Surface Condenser. Ninth Edition, 1300

Summer Avenue, Cleveland, Ohio 44115-2851, USA.

3. Introduction to Fluid Mechanics, Fifth Edition, Robert W. Fox & Alan T McDonald. John Wiley

& Sons, Inc.

4. Engineering Drawings of Steam Surface Condenser of Thermal Power Station.

5. An Introduction to Computational Fluid Mechanics: The Finite Volume Method, H.K. Versteeg

and W. Malalasekera.

6. Engineering fluid mechanics, R.J. Garde, A.G. Mirajgaoker, 1983, Nem Chand & Bros., Roorkee.