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物理方面的留学生课程作业范文指导-Pipe Flow and Head Loss due to Friction
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Introduction
There are two different head losses when water flow though a pipe. One is static head loss, which is caused by elevation. The other one is dynamic head loss, since water will lose potential energy due to the friction between pipe and water. It is a potential energy lost, instead of dynamic energy lost, since the flow rate is identical alone the pipe. There are basically two different types of flow, laminar flow and turbulent flow, which can be determined by Reynold’s number.
当水流通过管道有两种不同的水头损失。一个是静态的水头损失，这是造成高程。另一种是动态的水头损失，因为水会失去由于管道与水之间的摩擦的潜在能量。这是一个潜在的能量损失，而不是动态的能量损失，由于流量相同的单独的管。基本上有两种不同类型的流，层流和湍流，可由雷诺数的确定。
As McKEON AND SHARMA (2010) pointed out, ‘The accurate descriptions of statistical scaling and instantaneous structural coherence of turbulent fluctuations, and their relationship to the mean flow, are amongst the important unsolved problems in physics. Even at the simplest level, we still lack a complete explanation of the development of the mean velocity profile in canonical flows as the Reynold’s number increases.’
The first part of the experiment was done to found out the relationships between static head loss and elevation. The second part of the experiment was also done to find out the roughness, represented as friction factor, of a smooth pipe in both laminar flow and turbulent flow. The experiment results were compared with theoretical values. Exact values were hard to be predicted when Reynold’s number is large during the dynamic head loss test.
Methods
Static head loss test and dynamic head loss test were done in this experiment.
Static Head Loss Test:
The flow started at a high flow rate. The free jet, P, at each tapping was measured (Image 1). The pipe was titled and the free jet P’ at each tap was measured again. The height of each tap above the horizontal, Z, was also measured (Image 2).
Image 1: Observation of the free jet at tapping 3 Image 2: Static head loss in tilted pipe.
Dynamic head loss Test
A manometer tube was connected to three taps on a flow pipe (Image 3). This is read directly to give the head at each tapping. The manometer reads the head. (Head is the height equivalent to pressure per unit mass) To read the head for the laminar flow use the micrometer insert to work out the head loss.
Image 3: Diagram of flow in pipe with three taps connected to a manometer.
The experiment will be conducted 8 times in total, twice at each different flow rate:
Measure and record the distances between the taps on the pipe, L1-2 and L2-3 (refer to Image 3 above). Also record the internal diameter of the pipe. Take the water temperature and use the appropriate graph to determine, the kinematic viscosity (m2/s). Bleed the manometer by opening all 4 valves at once and then close all valves.
The following steps were repeated for each trial: The collection box was drained. Close the drain. Set the balance to zero and start the stopwatch at the same time. After 20kg water was collected, the stopwatch was stopped. Above steps were repeated twice more to get 3 readings. Open each manometer valve separately to observe the head at positions 1, 2 and 3 and record as h1, h2 and h3.
When going down from high flow rate to medium flow rate, leave tap 1 open and reduce the flow so that the head drops about 30%. Apply a similar reduction when going down from medium to low flow rate. For the laminar flow decrease the flow rate and attach the nozzle to the end of the pipe. Use the measuring cylinder rather than the collection box and time how long it takes to fill the cylinder. To determine fFDS, the theoretical friction factor read from the Moody chart, use the curve for smooth pipes.
Results
D, Internal diameter of pipe (m)
= 0.0162
±0.00005 A, Area of pipe (m2)
= 206x10-6
±0.6% r, Density of water (kg/m3)
= 998 L1-2 Length of section 1 to 2 (m)
= 0.990
±0.00005 L2-3 Length of section 2 to 3 (m)
= 0.990
±0.00005 T water temp (°C)
= 20 Kinematic Viscosity (m2/s)
= 1.00x10-6
±0.5%
Table 1: Basic data of pipe dimension, water temperature and kinematic viscosity
Table 2: Static Head Loss raw observations and calculations
Tap 1 Tap 2 Tap 3
Free Jet, P (m) P1 = 0.310±0.0005 P2 = 0.135±0.0005 P3 = 0.045±0.0005
Free Jet when tilted, P’ (m) P1’ = 0.175±0.0005 P2’ = 0.075±0.0005 P3’ = 0.035±0.0005
Height of tap, Z (m) Z1 = 0.23±0.01 Z2 = 0.06±0.01 Z3 = 0.01±0.01
P” = P’ + Z (m) P”1 = 0.31±0.01 P”2 = 0.14±0.01 P”3 = 0.05±0.01
Note: When height of tap, Z, was measured, ruler was used as horizontal zero which may contains large human error, so the error is estimated to be 0.01m.
#p#分页标题#e#
Figure 1: Static Head Loss, Head vs. Distance
Table 3: Dynamic Head Loss raw observations and calculations
M
mass of water volume of water Ave. flow time
tf Flow Rate
h1 h2 h3 Head
Difference 1
hf1=
h1-h2 Head
Difference 2
hf2=
h2-h3 Friction Factor
Using Equ A. Reynolds Number
Friction factor from MC
fFDS
f1 f2
(kg) (m3) (s) (x10-6
m3/s) (m) (m) (m) (m) (m) - - - -
High Flow Rate 1 20.0 0.0200 47.0 425 0.850 0.672 0.500 0.178 0.172 0.0134 0.0130 33422 0.0225
2 20.0 0.0200 44.7 447 0.934 0.735 0.540 0.199 0.195 0.0136 0.0133 35152 0.0227
Medium Flow Rate 1 20.0 0.0200 62.2 322 0.685 0.489 0.295 0.196 0.194 0.0257 0.0255 25322 0.0238
2 20.0 0.0200 61.5 325 0.694 0.492 0.297 0.202 0.195 0.0260 0.0251 25558 0.0237
Low Flow Rate 1 20.0 0.0200 83.3 240 0.484 0.367 0.254 0.117 0.113 0.0276 0.0267 18874 0.0265
2 20.0 0.0200 87.5 229 0.458 0.352 0.248 0.106 0.104 0.0275 0.0270 18009 0.0268
Laminar Flow 1 - 410
x10-6 22.9 17.9 0.216 0.215 0.214 0.001 0.001 0.0425 0.0425 1408 0.0455
2 - 410
x10-6 17.2 23.8 0.227 0.225 0.223 0.002 0.002 0.0240 0.0481 1872 0.0342
Error Analysis for Dynamic Head Loss. For high flow rate test 1:
M = 20.0 ±0.05 kg V = 0.0200±0.00005 m3
tf = 47.0±0.17 s #p#分页标题#e#Q = (425±0.61%) x10-6 m3/s
hf1 = 0.178±0.001 m f1 = 0.0134±3.3%
fFDS = 0.0225±0.0005
Figure 2: Experimental and Theoretical Friction Factor vs. Flow Rate
Figure 3: Experimental and Theoretical Friction Factor vs. Reynolds Number on Moody Chart
Absolute Roughness, e = Relative Roughness x Diameter
= 0.0002 x 0.0162 =
3.2 x 10-6m
Discussion
Static Head Loss
For static head loss, it can be see clearly from figure 1 that the free jet when tilted plus the height of tap is almost exactly the same as free jet when the pipe is horizontal. This proves that the static head loss at the same distance from entrance of pipe is caused by elevation of the pipe.
On the other hand, the head loss between tap 1 and tap 2 and the head lose between tap 2 and tap 3 is different in both horizontal and tilted pipes. This head loss should be caused by friction and also influenced by the free end of the pipe. This head loss should be considered as dynamic head loss. There are two different types of streamlines for the free end and the free jet. The distance between free end and free jet and flow rate determine the friction which, in turn, affects the pressure distribution. It will strike a balance and the head loss alone the pipe is not linear. However, the exact relationship need further experiment to be found out.
There could be large error, approximately 0.01m, when measuring the height of the tap when titled. Since the two distances between the three taps are same, the differences between the heights of the three taps when titled should also be same. In experiment, however, the height differences are 0.17m and 0.05m which suggest that the pipe is not perfect straight. This may causes errors when measuring the actual height free jet in static head loss test.
Dynamic Head Loss
The pipe is assumed to be smooth pipe. Figure 2 use log axis for flow rate which is linear to Reynolds Number for the same pipe, so the shapes of the curves are similar to the shape of the smooth pipe curve in Moody Chart. Figure 2 clearly shows that the experimental results of Laminar flow, Low Flow Rate and High Flow Rate are pretty close (in error bar) to the theoretical friction values. This proves that when the flow rate is not high (Reynolds Number is less than approximate 30000), the dynamic head loss is caused by the friction between water and pipe. The High Flow Rate tests (Reynold’s Number is greater than approximate 30000), however, gave much smaller friction values than theoretical ones. It is a big sudden drop which should be caused by other factors, such as significant turbulence in flow. It can be seem from figure 3 that the assumption of smooth pipe may not be valid, because the curve of relative roughness of 0.0002 could be chosen to give more accurate predicts.
Absolute roughness depends on the type and age of the material of pipe, while relative roughness also depends on the diameter of the pipe. In assumption, the pipe is smooth pipe which has zero absolute and relative roughness. In theory, a new drawn copper pipe should have absolute roughness, e, of 1.5 x 10-6m. In experiment, the absolute roughness of pipe is 3.2 x 10-6m, which indicates that the pipe is an old pipe.
For a particular pipe, Re is linear to the flow rate. Laminar flow occurs when a fluid flows in parallel layers, with no disruption between the layers (Batchelor, 2000). Laminar flow usually has low flow rate which gives low Re. Turbulent flow usually occurs when flow rate is high and has many unsteady vortices which appear on many scales and interact with each other. For pipe flow, a Reynolds number above about 4000 will most likely correspond to turbulent flow, while a Reynold's number below 2100 indicates laminar flow. The region in between (2100 < Re < 4000) is called the transition region (Fox, Robert W. 2009). In laminar flow, f=64/Re is used to predict the friction factor, while in turbulent flow, the curves in the Moody chart are used to predict the friction factor.
In this experiment, there is no test fall in the transition zone where more and more parts of the flow begin to change from laminar flow to turbulent as the flow rate increases. In transition zone, the linear interpolation could be used to predict the Re, although there could be large error.
When measuring the head at three tap, there is sometimes few air bubbles in the plastic pipe. This may causes error when reading the height. In the two high flow rate tests, 40kg water should be collected, instead of 20kg, to give more accurate flow rate.
Conclusion
Tow experiments were done to find what factors cause the static head loss and dynamic head loss. Static head loss is caused by elevation of the pipe. Dynamic head loss is caused by friction and should be considered as potential energy loss of flow. The assumption that the pipe is smooth is not valid. The copper pipe has relative roughness of 0.0002 and absolute roughness of 3.2 x 10-6m. The pipe is old while a new drawn copper pipe should have absolute roughness, e, of 1.5 x 10-6m. When Reynold’s number is less than 2100, the flow is laminar and experimental results agree with the theoretical result. When Reynold’s number is greater than 4000, the flow becomes turbulent, but the curve of relative roughness 0.0002 in Moody Chart still provides accurate predicts of friction factor until Reynold’s number exceeds 30000. When the flow rate is very high, Reynold’s number greater than 30000, the experiment results gives much smaller friction factor than that in theory. The reason could be the significant turbulence in the flow which makes the assumption of parabolic curve of speed distribution not valid.
Reference
Batchelor, G.K. 1973. An Introduction to Fluid Dynamics. Cambridge University Press
Fox, Robert W. 2009. www.ukassignment.org Introduction to fluid mechanics. 7th Edition. John Wiley. Hoboken, N.J.
McKEON, B. J. A. S. SHARMA and J. Fluid Mech(Editor). 2010. A critical-layer framework for turbulent pipe flow. vol. 658, pp. 336–382. Cambridge University Press
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