Zhiyong Dong
College of Civil Engineering at
Zhejiang University of Technology, Hangzhou 310014, Phone: 86-571-8320124, Fax:
86-571-8320124, E-mail: dongzy@mail.hz.zj.cn
Qinghua Xu
Nanjing Hydraulic Research
Institute, Nanjing 210029
Abstract: This paper presents an experimental study of local resistance coefficient on gooseneck-like pipe. The experimental results show the resistance coefficient of gooseneck-like pipe can not be simply calculated by use of linear superposition principle, it should be obtained from the intrinsic resistance law. Empirical formulae of gooseneck-like pipe are developed. And a criterion suitable to linear superposition calculation is given.
Keywords: gooseneck-like pipe, local resistance coefficient, linear superposition calculation, empirical formulae
A gooseneck-like pipe is usually met in hydraulic engineering as shown in Fig.1. In addition, it finds a widely application in water-supply and sewerage works, gas conveying pipelines, chemical engineering, cooling water circulating systems at thermal plant and nuclear power station etc. It is therefore significant to investigate hydraulic characteristics of gooseneck-like pipe (conduit).
This kind of gooseneck-like pipe generally consists of two elbows and a transition section. Many scholars such as Idelichik（1957）、Schlichting(1979) once investigated hydraulic characteristics of elbow in detail. However, the investigation of gooseneck-like pipe is less [3-5]. Engineering designers usually determine resistance coefficient of gooseneck-like pipe according to linear superposition principle, that is, the sum of resistance coefficient between two elbows and transition section, because of lack of investigation results of gooseneck-like hydraulic characteristics. For example, gooseneck-like pipe(culvert) in lock filling and emptying systems, i.e. No.1 Lock of Gezhouba Hydropower Station in Yangtze River, and Guiping Lock in Xijiang River in Guangxi Province, through a comparison of linear superposition calculation and measured values, we can find the relative error is 109% for the former, and up to 125% for the latter. It is obviously that the linear superposition values artificially increase resistance coefficient of gooseneck-like pipe, and considerably decrease discharge coefficient in lock filling and emptying systems, so it is difficult to bring out the systems latent potentiality, as a result of decrease of engineering efficiency. The reason of discrepancy between linear superposition calculation and measured values is that the flow in gooseneck-like pipe has its intrinsic hydraulic characteristics.
Fig.1 Gooseneck-like pipe
The experimental model is square pipe. The
cross-sectional dimension is 15×15cm.
The working section is made of Perspex, the conveying section made of plastic
plate. The control valve is installed in the end of conveying pipe. The lower
discharging flume measures the flow-rate. The geometric parameters of
gooseneck-like pipe are the centered angle of elbow
and
, the relative curvature radius
and 2.5, relative transition length
, where
is gradually uniform length from upper elbow outflow. The entrance section of
gooseneck-like pipe is much longer so as to meet the requirement of flow
established.
The experiment has conducted under steady
flows. The experimental flow-rate is
，Reynolds number
，（
，d denotes pipe
diameter），The
flow in pipe is fully developed turbulent boundary layer. The longitudinal
velocity distribution is measured with small Pitot tube and Pitot cylinder,
respectively. The three-dimensional velocity and cross-sectional pressure are
measured with Pitot sphere.
Comparisons of linear superposition calculation
and measured values of several typical resistance coefficients of gooseneck-like
pipe are shown in Fig.2-4. The abscissa
and ordinate
in the figures is relative transition length and local resistance coefficient of
gooseneck-like pipe, respectively. It follows from the figures that linear
superposition value is always greater than measured one, and the discrepancy is
considerable, when transition length of gooseneck-like pipe is less than gradual
uniform length of upper elbow. The reason is that gooseneck-like pipe change its
flow characteristics of geometric consistent parts. Hence resistance coefficient
of gooseneck-like pipe can not be calculated by use of linear superposition
method when the transition length is greater than or equal gradual uniform
length of upper elbow outflow, whose evaluation criterion will be discussed as
below.
---linear superposition, —the authors’ formulae,
○measured data from the authors, △reference[4]
Fig.
2 
relation when
---linear superposition, —the authors’ formulae,
○measured data from the authors, △reference[4]
Fig.
3 
relation when
---linear superposition, —the authors’ formulae,
○measured data from the authors, △reference[4], □reference[5]
Fig.
4 
relation when
We develop empirical formulae of local resistance coefficient for gooseneck-like pipe based on dimensional analysis and π theorem from our experimental data as follows:
If relative transition length
，resistance
coefficient of gooseneck-like pipe can be calculated in the following formulae:
（1）
If relative transition
length
，we have
（2）
If relative transition length
，resistance
coefficient of gooseneck-like pipe can be calculated in the following formulae:
（3）
If relative transition
length
，we have
（4）
in which
denotes relative transition length
of gooseneck-like pipe;
,
is centered angle and relative
curvature radius of upper and lower elbows in the gooseneck-like pipe,
respectively;
The above formulae are
suitable for
，
，
.
It should be noted that equations（1）～（4）are not only for rectangular cross-section, but also for circular, arch and other forms. It is required to replace pipe diameter d by hydraulic radius R.
Only when
, can we calculate resistance coefficient of gooseneck-like pipe according to
linear superposition principle. It is easily to obtain
. However, the reference [3] indicates that just when relative transition length
of gooseneck-like pipe
, resistance coefficient
can be linearly superposed. It is obviously that the results of reference [3]
are different from our experimental ones.
We compare the calculated resistance coefficient of gooseneck-like culvert with the measured ones for No.1 and No.2 Locks at Gezhouba Hydropower Engineering in Yangtze River and Guiping Lock in Guangxi Province as shown in Table 1. It follows from the Table that the calculated values by formulae in this paper basically agree with measured ones.
Table 1 Comparison of calculated values by this paper’s formulae and measured ones
|
|
|
|
|
measured |
calculated |
|
No.1 Lock |
|
2.15 |
3.04 |
0.142 |
0.163 |
|
No.2 Lock |
|
1.95 |
4.96 |
0.159 |
0.154 |
|
Guiping Lock |
|
1.25 |
1.80 |
0.167 |
0.215 |
We can draw the following conclusions through experimental study of resistance coefficient of gooseneck-like pipe as follows:
(1) If transition length of gooseneck-like pipe is less than gradual uniform section from upper elbow outflow, the linear superposition method can not be applied to the resistance coefficient calculation, otherwise it can be led erroneous results.
(2) The empirical formulae developed in this paper can solve the problems of resistance calculation in engineering practice, and agree well with measured values.
(3) Only when transition length of
gooseneck-like pipe
×
, can we calculate the resistance coefficient according to linear superposition
method.
References
[1] I.E Idelichik：Hydraulic Resistance，Electric Power Industry Press, 1957.
[2] Schlichting, H.： Boundary-Layer Theory, McGraw-Hill Company, 7th edition, 1979.
[3] A.M. Kylkannov and N.F. Fijdlov：Handbook of Hydraulic Computation on Water Supply and Sewerage Systems, China Civil Engineering Press, 1983.
[4] Shaozeng Hua and Xuening Yang et al: Practical Handbook of Fluid Resistance，China National Defense Industry Press, 1985.
[5] Weixin Wang: Fluid Mechanics, Coal Industry Press, 1986.
