Fluid Friction Head Loss And Velocity Relationship In Smooth Pipes

Abstract

The main objective of this experiment was to determine the relationship between fluid head loss and velocity of water flowing through smooth pipes, and to compare the values of measured head loss and those obtained through calculation using friction equation of a pipe. This was achieved by obtaining a series of head loss readings at different flow rates of water flowing through varied smooth test pipes. The findings of the experiment showed that a graph of h vs. u for the different pipe sizes had three main zones: laminar zone, transition zone and turbulent zone. The graph for the laminar zone was a straight line (h = u). A graph of log h vs. log u for the different pipe sizes also had three main zones: laminar zone, transition zone and turbulent zone. The graph for the turbulent zone was a straight line (h = uⁿ) with a gradient of 1.9 and 1.86 (the values of n) for pipe 8 and pipe 10 respectively. The difference between measured head loss values and calculated head loss values was relatively small. This confirmed that head loss through a pipe can be predicted using friction equation of a pipe as long as pipe dimensions and velocity of fluid flowing through the pipe are known. Engineers can use findings from this experiment to design pipes so that water flows through them at optimal velocity so as to reduce head loss depending on whether the flow is laminar or turbulent. Generally, mean velocity has to be kept low so as to minimize head losses due to friction.

The aim of this experiment is to determine the relationship between fluid friction head loss and velocity of water flowing through smooth bore pipes, and to compare measured head loss values and those obtained through calculation using friction equation of a pipe.

Fluid friction head losses occur as a result of an incompressible fluid flow through pipe flow metering devices, valves, pipes and bends. The values of these losses are different in smooth and rough pipes, with the latter pipes have higher values than the former pipes (Soumerai & Soumerai-Bourke, 2014). In smooth pipes, it is possible to determine friction head losses over Reynold’s numbers ranging between 10³ and 10⁵. Within this range, the smooth pipes’ laminar flow, transitional flow and turbulent flow are covered. In rough pipes, friction head losses are determined at high Reynold’s numbers. Various pipe components, such as control valves and pipe fittings, also affects friction head losses.

According to Prof. Osborne Reynolds, the flow of fluid through a pipe can either be laminar flow or turbulent flow (Launder & Jackson, 2007). Laminar flow occurs when velocity of the fluid is low (Cerbus, et al., 2018), and the relationship between fluid friction head loss, h, and fluid velocity, u, is: . On the other hand, turbulent flow occurs when velocity of the fluid is higher (Hanjalic & Launder, 2011); (Jackson & Launder, 2011), and the relationship between fluid friction head loss and fluid velocity is: .  There is a phase known as transition phase that separates the laminar flow and turbulent flow (Launder, 2015); (Wu, et al., 2015). In the transition flow or phase, h and u do not have any definite relationship (Tribonet, 2018).

Aim of Experiment

The formulae for determining friction head loss and Reynold’s number are provided in equation 1 and 2 below

Where L = length of pipe from one tapping to another, u = mean velocity of fluid (or water) flowing through the pipe (m/s), d = pipe’s internal diameter, f = friction coefficient of pipe, g = gravitational acceleration (m/s²), ρ = density (999 kg/m³ at 15°C) and μ = molecular viscosity (1.15 x 10^-3 Ns/m² at 15°C). λ is also equivalent to 4f.

Determining fluid friction head losses helps engineers to estimate the amount of energy lost due to friction when a fluid is flowing through a pipe (Nuclear Power, 2018).

The apparatus used in this experiment is Armfield C6-MKII-10 Fluid Friction Apparatus together with Armfield F1-10 Hydraulics Bench. Other devices used are internal vernier caliper and stop watch. The pipes used in this experiment are assumed to have constant internal diameters.

The pipe network (as shown in Appendix 1) was primed with water. Appropriate valves were opened and closed so as to obtain the required water flow through the right test pipe. Readings were taken at different flow rates, with the flow being changed using control valves fitted on the hydraulics bench. Volumetric tank or measuring cylinder was used to measure flow rates. Head loss from one tapping to another was also measured using pressurized water manometer or portable pressure meter. Readings on all the four smooth test pipes were obtained and recorded. Internal diameter of the test pipe samples was also measured.

Data, Results and Graphs

Graphs of h vs. u for the different pipes sizes are as follows:

The values of h were measured from the experiment. However, the values of u are calculated using the equation:  (where Q = flow rate through the test pipe (m³/s) and d = internal diameter of pipe (m).

Graph of h vs. u for pipe 8

In this test, d = 17.2mm = 0.0172m

Sample calculation of u for the first value of Q for pipe 8 is as follows

 = 3.7354 m/s  

Calculated values of h are obtained using equation 1 where f = 0.015, L = 1m, g = 10 m/s² and d = 0.0172 m

Table 1 below shows the Q, u and h data for pipe 8:

Table 1: Experimental data for pipe 8

Flow Rate Q (m3/sec)	Measured Head loss (m)	Calculated head loss (m)	Mean velocity, (m/s)
Q	h	h	u
8.679E-04	1.10676	2.434E+00	3.735E+00
7.667E-04	0.736	1.899E+00	3.300E+00
7.041E-04	0.65504	1.602E+00	3.030E+00
6.586E-04	0.6118	1.401E+00	2.834E+00
6.571E-04	0.51244	1.395E+00	2.828E+00
5.247E-04	0.41676	8.895E-01	2.258E+00
5.107E-04	0.27232	8.426E-01	2.198E+00
4.293E-04	0.28612	5.955E-01	1.848E+00
3.987E-04	0.24656	5.135E-01	1.716E+00
3.680E-04	0.276	4.375E-01	1.584E+00
3.373E-04	0.19044	3.676E-01	1.452E+00
3.067E-04	0.15732	3.038E-01	1.320E+00
2.760E-04	0.12696	2.461E-01	1.188E+00
2.453E-04	0.10028	1.945E-01	1.056E+00
2.147E-04	0.07728	1.489E-01	9.239E-01
1.995E-04	0.05336	1.286E-01	8.585E-01
1.840E-04	0.0644	1.094E-01	7.919E-01
1.533E-04	0.04508	7.596E-02	6.599E-01
1.380E-04	0.03864	6.153E-02	5.939E-01
1.227E-04	0.03036	4.861E-02	5.279E-01
1.073E-04	0.02484	3.722E-02	4.619E-01
9.782E-05	0.0184	3.091E-02	4.210E-01
9.200E-05	0.0184	2.734E-02	3.960E-01
7.667E-05	0.01288	1.899E-02	3.300E-01
6.900E-05	0.01104	1.538E-02	2.970E-01
6.133E-05	0.0092	1.215E-02	2.640E-01
5.367E-05	0.00644	9.305E-03	2.310E-01
4.600E-05	0.002576	6.836E-03	1.980E-01
3.833E-05	0.001748	4.747E-03	1.650E-01
3.462E-05	0.002116	3.872E-03	1.490E-01
3.067E-05	0.00184	3.038E-03	1.320E-01
2.668E-05	0.001656	2.300E-03	1.148E-01
2.300E-05	0.001196	1.709E-03	9.899E-02
1.533E-05	0.00092	7.596E-04	6.599E-02
1.472E-05	0.03772	7.000E-04	6.335E-02
1.454E-05	0.00092	6.831E-04	6.258E-02
1.380E-05	0.00092	6.153E-04	5.939E-02
1.288E-05	0.00828	5.360E-04	5.543E-02
1.233E-05	0.00368	4.910E-04	5.306E-02
1.227E-05	0.00092	4.861E-04	5.279E-02
9.024E-06	0.00552	2.631E-04	3.884E-02
9.091E-05	0.091	2.670E-02	3.913E-01
4.348E-05	0.01	6.107E-03	1.871E-01
4.193E-05	0.024	5.681E-03	1.805E-01
2.827E-05	0.046	2.581E-03	1.217E-01
1.083E-05	0.005	3.792E-04	4.662E-02

The graph of h (measured) vs. u for pipe 8 is as shown in Figure 1 below

From Figure 1 above, the graph for the laminar flow zone is a straight line. This ascertains the relationship

Graph of calculated head vs. mean velocity for pipe 8 is as shown in Figure 2 below

The graphs in Figure 1 and 2 above are similar. This shows that the error in the experiment is small and that the equation 1 can also be used to predict the values of h.

Graph of h vs. u for pipe 10

In this test, d = 7.7mm = 0.0077m

Sample calculation of u for the first value of Q for pipe 10 is as follows

Brief Introduction/Background

 = 0.13314 m/s  

Table 2 below shows the Q, u and h data for pipe 10

Table 2: Experimental data for pipe 10

Flow Rate Q (m3/sec)	Measured Head loss (m)	Calculated head loss (m)
Q	h	h	Mean velocity
an the client give	0		u
6.200E-06	7.943	6.907E-03	1.331E-01
8.400E-06	8.08	1.268E-02	1.804E-01
9.200E-06	0.029	1.521E-02	1.976E-01
1.500E-05	7.987	4.043E-02	3.221E-01
1.540E-05	8.041	4.261E-02	3.307E-01
1.913E-05	0.06	6.578E-02	4.109E-01
2.400E-05	0.67	1.035E-01	5.154E-01
2.476E-05	8.176	1.101E-01	5.316E-01
4.150E-05	0.15	3.094E-01	8.912E-01
4.667E-05	8.257	3.913E-01	1.002E+00
5.807E-05	0.202	6.058E-01	1.247E+00
6.780E-05	0.73	8.259E-01	1.456E+00
8.333E-05	0.873	1.248E+00	1.790E+00
1.073E-04	1.288	2.070E+00	2.305E+00
1.095E-04	1.61	2.154E+00	2.351E+00
1.111E-04	2.317	2.218E+00	2.386E+00
1.230E-04	1.61	2.718E+00	2.641E+00
1.342E-04	2.0125	3.234E+00	2.881E+00
1.346E-04	1.61	3.254E+00	2.890E+00
1.610E-04	2.8175	4.657E+00	3.457E+00
1.610E-04	2.737	4.657E+00	3.457E+00
1.857E-04	3.9445	6.196E+00	3.988E+00
1.878E-04	3.7835	6.339E+00	4.034E+00
1.987E-04	8.363	7.096E+00	4.268E+00
2.000E-04	4.871	7.187E+00	4.295E+00
2.013E-04	3.703	7.277E+00	4.322E+00
2.013E-04	4.508	7.277E+00	4.322E+00
2.143E-04	4.83	8.253E+00	4.603E+00
2.147E-04	4.7495	8.280E+00	4.610E+00
2.292E-04	5.635	9.440E+00	4.922E+00
2.308E-04	5.635	9.570E+00	4.956E+00
2.415E-04	5.957	1.048E+01	5.186E+00
2.632E-04	5.046	1.244E+01	5.651E+00
2.683E-04	7.1645	1.294E+01	5.762E+00
2.683E-04	7.6475	1.294E+01	5.762E+00
2.683E-04	6.8425	1.294E+01	5.762E+00
2.683E-04	7.728	1.294E+01	5.762E+00
2.683E-04	6.923	1.294E+01	5.762E+00
2.683E-04	7.245	1.294E+01	5.762E+00
2.800E-04	8.05	1.409E+01	6.013E+00
2.952E-04	8.694	1.565E+01	6.339E+00
3.052E-04	10.0625	1.673E+01	6.553E+00
3.067E-04	9.66	1.690E+01	6.586E+00
3.086E-04	8.4525	1.711E+01	6.628E+00
3.212E-04	10.465	1.854E+01	6.898E+00
3.220E-04	10.2235	1.863E+01	6.915E+00
3.220E-04	10.465	1.863E+01	6.915E+00
3.240E-04	4.974	1.887E+01	6.959E+00
3.251E-04	10.465	1.899E+01	6.982E+00
3.389E-04	11.27	2.064E+01	7.279E+00
3.433E-04	11.6725	2.117E+01	7.372E+00
3.488E-04	11.9945	2.186E+01	7.491E+00
3.541E-04	8.42	2.253E+01	7.604E+00
3.757E-04	13.685	2.536E+01	8.067E+00
3.808E-04	5.977	2.606E+01	8.178E+00
4.025E-04	15.8585	2.911E+01	8.644E+00
4.025E-04	15.456	2.911E+01	8.644E+00
4.025E-04	15.295	2.911E+01	8.644E+00
4.049E-04	15.134	2.946E+01	8.696E+00
4.112E-04	7.325	3.038E+01	8.830E+00
4.136E-04	7.87	3.073E+01	8.881E+00
4.136E-04	8.107	3.073E+01	8.881E+00

The graph of h vs. u for pipe 10 is as shown in Figure 3 below

From Figure 2 above, the graph for the laminar flow zone is relatively a straight line. This ascertains the relationship

Graph of calculated head vs. mean velocity for pipe 10 is as shown in Figure 4 below

The graphs in Figure 3 and 4 above are similar. This shows that equation 1 can be used to predict the values of h and that errors in the experiment were small thus negligible.

Graph of log h vs. log u for pipe 8

Table 3 below shows log u and log h values for pipe 8

Table 3: Experimental log u and log h (measured and calculated) data for pipe 8

log u	Log h (measured)	Log h (calculated)
0.572335	0.044053	0.386263
0.51846	-0.13312	0.278512
0.481476	-0.18373	0.204545
0.452445	-0.21339	0.146482
0.451513	-0.29036	0.144619
0.353777	-0.38011	-0.05085
0.342007	-0.56492	-0.07439
0.266648	-0.54345	-0.22511
0.234463	-0.60808	-0.28948
0.199701	-0.55909	-0.35901
0.161912	-0.72024	-0.43458
0.12052	-0.80322	-0.51737
0.074762	-0.89633	-0.60888
0.02361	-0.99879	-0.71119
-0.03438	-1.11193	-0.82717
-0.06625	-1.27278	-0.8909
-0.10133	-1.19111	-0.96107
-0.18051	-1.34602	-1.11943
-0.22627	-1.41296	-1.21094
-0.27742	-1.5177	-1.31325
-0.33541	-1.60485	-1.42923
-0.37572	-1.73518	-1.50984
-0.40236	-1.73518	-1.56313
-0.48154	-1.89008	-1.72149
-0.5273	-1.95703	-1.813
-0.57845	-2.03621	-1.91531
-0.63644	-2.19111	-2.03129
-0.70339	-2.58905	-2.16519
-0.78257	-2.75746	-2.32355
-0.8268	-2.67448	-2.41202
-0.87948	-2.73518	-2.51737
-0.93996	-2.78094	-2.63833
-1.00442	-2.92227	-2.76725
-1.18051	-3.03621	-3.11943
-1.19824	-1.42343	-3.15489
-1.20356	-3.03621	-3.16552
-1.22627	-3.03621	-3.21094
-1.25623	-2.08197	-3.27087
-1.27525	-2.43415	-3.30892
-1.27742	-3.03621	-3.31325
-1.41074	-2.25806	-3.57989
-0.40754	-1.04096	-1.57349
-0.72787	-2	-2.21416
-0.74359	-1.61979	-2.24558
-0.91487	-1.33724	-2.58815
-1.33138	-2.30103	-3.42118

The graph of log h vs. log u for pipe 8 is as shown in Figure 5 below

From Figure 5 above, the graph for the turbulent flow zone is a straight line. This ascertains the relationship. The slope of the straight line is determined as follows:

(0.5723355173, 0.044033455) and (0.023609771, -0.998785675)

Gradient = change in y/change in x =

Therefore the value of n = 1.9

The straight line equation for the turbulent flow zone is given by h = uⁿ. This can be rewritten as follows:

Log h = log uⁿ 

Using the laws of logarithm, the above equation can be written as: log h = n log u. Therefore if a graph of log h vs. log u is plotted, the gradient of that straight line represents the value of n.

From Figure 5 above, the graph for the turbulent flow zone is a straight line. This ascertains the relationship.

The graphs in Figure 5 and 6 are similar. This shows that equation 1 can be used to predict the

values of head loss (h).

Graph log h vs. log u for pipe 10

Table 4 below shows log u and log h values for pipe 10

Table 4: Experimental log u and log h data for pipe 10

Log u	Log h (measured)	Log h (calculated)
-8.757E-01	9.000E-01	-2.50977
-7.438E-01	9.074E-01	-2.24599
-7.043E-01	-1.538E+00	-2.16697
-4.920E-01	9.024E-01	-1.74237
-4.806E-01	9.053E-01	-0.75841
-3.863E-01	-1.222E+00	-1.53097
-2.879E-01	-1.739E-01	-1.33413
-2.744E-01	9.125E-01	-1.3072
-5.002E-02	-8.239E-01	-0.85845
9.355E-04	9.168E-01	-0.75654
9.586E-02	-6.946E-01	-0.5667
1.632E-01	-1.367E-01	-0.43209
2.527E-01	-5.899E-02	-0.25291
3.627E-01	1.099E-01	-0.03308
3.713E-01	2.068E-01	-0.01588
3.777E-01	3.649E-01	-0.00303
4.218E-01	2.068E-01	0.085223
4.596E-01	3.037E-01	0.16074
4.609E-01	2.068E-01	0.163349
5.388E-01	4.499E-01	0.319102
5.388E-01	4.373E-01	0.319102
6.007E-01	5.960E-01	0.443064
6.057E-01	5.779E-01	0.452996
6.302E-01	9.224E-01	0.501969
6.330E-01	6.876E-01	0.50751
6.357E-01	5.686E-01	0.512922
6.357E-01	6.540E-01	0.512922
6.630E-01	6.839E-01	0.567591
6.637E-01	6.766E-01	0.568979
6.922E-01	7.509E-01	0.625933
6.951E-01	7.509E-01	0.631889
7.148E-01	7.750E-01	0.671285
7.521E-01	7.029E-01	0.745883
7.606E-01	8.552E-01	0.7628
7.606E-01	8.835E-01	0.7628
7.606E-01	8.352E-01	0.7628
7.606E-01	8.881E-01	0.7628
7.606E-01	8.403E-01	0.7628
7.606E-01	8.600E-01	0.7628
7.791E-01	9.058E-01	0.799766
8.020E-01	9.392E-01	0.845585
8.164E-01	1.003E+00	0.874492
8.186E-01	9.850E-01	0.878783
8.214E-01	9.270E-01	0.884316
8.387E-01	1.020E+00	0.91908
8.398E-01	1.010E+00	0.921162
8.398E-01	1.020E+00	0.921162
8.425E-01	6.967E-01	0.926658
8.440E-01	1.020E+00	0.929541
8.621E-01	1.052E+00	0.965715
8.676E-01	1.067E+00	0.976756
8.745E-01	1.079E+00	0.990686
8.811E-01	9.253E-01	1.003721
9.067E-01	1.136E+00	1.055056
9.126E-01	7.765E-01	1.066861
9.367E-01	1.200E+00	1.114982
9.367E-01	1.189E+00	1.114982
9.367E-01	1.185E+00	1.114982
9.393E-01	1.180E+00	1.120209
9.460E-01	8.648E-01	1.133523
9.485E-01	8.960E-01	1.138538
9.485E-01	9.089E-01	1.138538

The graph of log h vs. log u for pipe 10 is as shown in Figure 7 below

(0.9367, 1.189) and (0.5388, 0.4499)

Gradient = change in y/change in x =

Therefore the value of n = 1.86

Graph of log h (calculated vs. log u for pipe 10

The graphs in This shows that equation 1 can be used to predict the

values of head loss (h).

Estimation of Reynolds number (Re) at the start and finish of transition phase

Re is calculated using equation 2

For pipe 8:

Re at the start of transition phase

At the start of transition phase, ρ = 999 kg/m³, d = 0.0172m, μ = 1.15 x 10^-4 kgs/m² and u = 1.452 m/s

Using equation 2, Re is calculated as follows:

 = 21,695.2

Re at the start of transition phase

At the end of transition phase, ρ = 999 kg/m³, d = 0.0172m, μ = 1.15 x 10^-4 kgs/m² and u = 2.834 m/s.

Description of Apparatus

Using equation 2, Re is calculated as follows:

 = 42,344.4

For pipe 10:

Re at the start of transition phase

At the start of transition phase, ρ = 999 kg/m³, d = 0.0077m, μ = 1.15 x 10^-4 kgs/m² and u = 4.034 m/s

Using equation 2, Re is calculated as follows:

 = 26,983.3

Re at the start of transition phase

At the end of transition phase, ρ = 999 kg/m³, d = 0.0172m, μ = 1.15 x 10^-4 kgs/m² and u = 5.762 m/s.

Using equation 2, Re is calculated as follows:

 = 46,254.1

Comparison between calculated and measured head losses

For pipe 8:

Calculated head loss,  (where L = 1m, g = 10 m/s², d = 0.0172 m, λ = obtained from Moody’s diagram, and u = calculated values

u	Re	f	λ	Calculated head loss, he	Measured head loss, h
3.735	55806	0.005	0.02	1.39502	1.10676
0.04662	697	0.024	0.096	0.00607	0.005

For pipe 10:

u	Re	f	λ	Calculated head loss, he	Measured head loss, h
0.133	890	0.018	0.072	8.317	7.943
8.881	59404	0.0048	0.0192	9.833	8.107

From Figure 1 and 2 above, the laminar flow zone, transition flow zone and turbulent flow zone are identified by the vertical dotted lines. The first zone is the laminar flow zone where the graph of h vs. u is a straight line. This means that head loss in laminar flow is proportional to the mean velocity of water flowing through the pipe. The second zone is the transition flow zone where there is no definite relationship between h and u. The third zone is the turbulent flow zone where the graph of h vs. u is not perfectly linear.

The graphs in Figure 3 and 4 shows the relationship between log h and log u for pipe 8 and pipe 10. From these graphs, the turbulent flow zones show linear relationship between log h and log u. The straight lines of the graphs in the turbulent flow zone are represented by the equation log h = n log u (h = uⁿ). Therefore the gradients of these straight lines represent the values of n.   

Therefore the graph for the turbulent flow zone in Figure 3 above is a straight line. This ascertains the relationship. Also, the graph for the turbulent flow zone in Figure 4 above is a straight line. This ascertains the relationship. This means that head loss in turbulent flow is proportional to the square of mean velocity of water flowing through the pipe.

There was some small differences between the measured head loss values and calculated head loss values. These differences were probably due to possible sources of errors. One of the possible sources of errors was defective apparatus and devices. If any of the apparatus or devices used was defective then it generated a systematic error throughout the experiment. Another possible source of error is incorrect reading and recording of measurements. The errors may also have been due to unsteady temperature. The temperature of water used in this experiment was assumed to be 15°C. It is likely that this temperature was not maintained throughout the experiment resulting to errors. Last but not least, the errors may have been due to incorrect calculations (arithmetic errors). In the future, the experiment can be improved by ensuring that apparatus and devices used are not defective, all measurements are read and recorded correctly, temperature of the water used is maintained at the recommended value or the variations are considered in the calculations, and all arithmetic or calculations are done correctly.     

Methodology

Conclusion

This experiment demonstrated that was flowing through a rough pipe exhibits laminar flow, transition flow and turbulent flow. Each of these flows have unique head loss and mean velocity. Velocity of laminar flow is less than that of transition flow while that of turbulent flow is the greatest. The relationship between h and u for laminar flow is linear, and the relationship between h and uⁿ for turbulent flow is also linear. However, the relationship between h and u or h and uⁿ for transition flow was not definite. For pipe 8, the gradient (value of n) of the straight line of log h vs. log u graph was 1.9, while for pipe 10, the gradient (value of n) of the straight line of log h vs. log u graph was 1.86. The results obtained also showed small differences between measured and calculated values of head loss. Therefore head losses through a pipe can be obtained either by taking measurements or calculating using pipe friction equation, depending on which method is easier or the resources provided. From this experiment, it is evident that velocity of water flowing through a pipe has a direct impact on head loss. Low velocity results to a smaller head loss and vice versa. Therefore findings from this experiment can be used to design pipes so that water flows at optimal velocity depending on whether the flow is laminar or turbulent. To minimize head losses due to friction, mean velocity of water through pipes has to be kept low.     

Abstract

The objective of this experiment was to determine head loss of water flowing through typical fittings used in plumbing installations. This was achieved by measuring differential head from one tapping to another on test valves and fittings. From the experiment, it was found that head loss is proportional to the fluid’s mean velocity. A graph of fitting factor against flow rate showed that the fitting factor was constant for the 45° elbow and 90° elbow fittings. However, this was not the case for the isolating valve fitting. Therefore pipe fittings cause head loss making it important to select and use appropriate type of fittings for plumbing works so as to reduce head loss.

Aim of Experiment

The aim of this experiment is to determine head loss of water flowing through typical fittings that are used in plumbing installations.

Brief Introduction/Background

Pipe fittings experience head loss when water is flowing through them. The pipe fitting’s head loss is proportional to the fluid velocity head. This relationship is represented by equation 3 below

Where K = loss factor of the pipe fitting, u = water/fluid’s mean velocity (m/s) and g = gravitational acceleration (m/s²). It is important to note that control valves of flow are pipe fittings that have adjustable K factor.

The apparatus used in this experiment is Armfield C6-MKII-10 Fluid Friction Apparatus together with Armfield F1-10 Hydraulics Bench. Other devices used are internal vernier caliper and stop watch. The pipes used in this experiment are assumed to have constant internal diameters. The following valves and fittings were also provided for the experiment: sudden enlargement, sudden contraction, 45° Y junction, 45° mitre, 45° elbow, ball valve, in line strainer, globe valve, gate valve, 90° T junction, 90° long radius bend, 90° short radius band and 90° elbow.

Data, Results and Graphs

Methodology

The pipe network was connected in accordance with the diagram provided in the lab manual. After setting up the equipment, the pipe network was primed with water. Appropriate valves were opened and closed so as to obtain the required water flow through the desired fitting. Readings of volume and time were taken and recorded at different flow rates, with the control valve fixed on the hydraulics bench being used to adjust flow of the water. Volumetric tank was used to measure flow rates of the water. Pressurized water manometer, sensors or hand held pressure meter was used to measure differential head from one tapping to another on various fittings.

Data, Results and Graphs

The internal diameter of pipe 8, d = 17.2 mm = 0.0172 m, and g = 10 m/s² 

The values of u are calculated using the equation:  (Q is the measured flow rate from the experiment).

The values of hv are calculated using the equation:

The values of K are calculated using the equation: K

Table 5 below shows the experimental data for pipe 8

Table 5: Experimental data for pipe 8

Flow Q (m3/s)	Head loss elbow 45 (m)	Head loss isolating valve (m)	Head loss 90° (m)	u	hv	K (elbow)	K (isolating valve)	K (90)
7.590E-06	0.0161	0.00805	0.0069	3.267E-02	5.33531E-05	301.7629	150.8814684	129.327
1.128E-05	0.03105	0.0046	0.0069	4.855E-02	0.000117844	263.4833	39.03455762	58.55184
1.541E-05	0.0138	0.023	0.00805	6.632E-02	0.000219929	62.74764	104.5793967	36.60279
1.610E-05	0.01035	0.0184	0.01725	6.929E-02	0.000240065	43.11339	76.6460264	71.85565
1.818E-05	0.0023	0.01265	0.00115	7.823E-02	0.00030597	7.517081	41.34394441	3.75854
1.840E-05	0.0391	0.00345	0.0092	7.919E-02	0.000313554	124.6995	11.00289637	29.34106
3.335E-05	0.02645	0.03565	0.01265	1.435E-01	0.001030073	25.67779	34.60918968	12.28068
3.752E-05	0.228	0.544	0.52	1.615E-01	0.001303954	174.8528	417.1925397	398.787
6.499E-05	0.878	0.521	0.538	2.797E-01	0.003911218	224.4825	133.206572	137.553
7.494E-05	0.38	0.019	0.093	3.225E-01	0.00520121	73.05993	3.652996373	17.88046
9.518E-05	0.818	0.634	0.543	4.097E-01	0.008390766	97.48811	75.55924753	64.71399
1.223E-04	0.0046	0.00805	0.00345	5.263E-01	0.013846962	0.332203	0.581354948	0.249152
1.421E-04	0.0256	0.506	0.568	6.115E-01	0.018697257	1.369185	27.06279298	30.37879
1.827E-04	0.024	0.495	0.608	7.862E-01	0.030907711	0.776505	16.01542096	19.67147
1.827E-04	0.026	0.024	0.069	7.862E-01	0.030907711	0.841214	0.776505259	2.232453
2.493E-04	0.0414	0.0621	0.13455	1.073E+00	0.057582897	0.718963	1.078445221	2.336631
3.187E-04	0.119	0.099	0.22	1.372E+00	0.094052544	1.26525	1.052603104	2.339118
3.501E-04	0.311	0.045	0.279	1.507E+00	0.11354291	2.739053	0.396325936	2.457221
4.040E-04	0.223	0.179	0.36	1.739E+00	0.15114217	1.475432	1.184315403	2.381863
4.472E-04	0.153	0.207	0.442	1.925E+00	0.185239171	0.825959	1.117474232	2.386104
4.980E-04	0.4	0.82	0.54	2.143E+00	0.229693661	1.74145	3.569972271	2.350957
6.384E-04	0.23	0.3289	0.77395	2.747E+00	0.377401459	0.609431	0.871485768	2.050734
6.559E-04	0.3036	0.4301	0.9568	2.823E+00	0.398421832	0.762006	1.079509116	2.401475
8.214E-04	0.253	0.08625	0.12075	3.535E+00	0.62490793	0.40486	0.138020332	0.193228
8.232E-04	0.4025	0.5589	1.3225	3.543E+00	0.62759474	0.641337	0.890542837	2.107252
8.801E-04	0.44275	0.59225	1.4421	3.788E+00	0.717368797	0.617186	0.825586508	2.010263
9.583E-04	1.357	0.414	0.5865	4.124E+00	0.850569127	1.595402	0.486732926	0.689538

A graph of fitting factor (K) against flow rate (Q) for 45° elbow is as shown in Figure 5 below

A graph of fitting factor (K) against flow rate (Q) for isolating valve is as shown in Figure 6 below

A graph of fitting factor (K) against flow rate (Q) for 90° elbow is as shown in Figure 7 below

The graphs of fitting factor (K) versus flow rate (Q) in Figure 5 and 7 above shows that K is a constant for 45° elbow and 90° elbow past a certain value of flow rate. At the start (low flow rate), fitting factor decreased gradually with increasing flow rate. This is probably because when flow rate was starting to increase, the flow through the pipe was relatively stable hence the interaction between the water and internal surface of the pipe was regulated. As the flow rate continued to increase, the interaction between water and internal pipe surface became normalized and that is why the fitting factor became constant (Burger, et al., 2010). However, this was not the case for isolating valve (as shown by the graph in Figure 6 above) where K is not constant. This shows that for 45° elbow and 90 °elbow pipe fittings, head loss remains constant regardless of the flow rate of water through the pipe fitting.    

Conclusion

This experiment confirmed that head loss is proportional to the fluid’s mean velocity. A graph of fitting factor against flow rate showed that fitting factor was constant for the 45° elbow and 90° elbow fittings even as flow rate increased. However, the trend was different for the isolating valve fitting because fitting factor was not constant. This experiment showed that pipe fittings cause head loss making it important to select and use appropriate type of fittings for plumbing works so as to reduce head loss.

Experiment C: Fluid friction in a Rough Bore Pipe

The objective of this experiment was to determine the relationship between coefficient of friction and Reynolds’ number of water flowing through a rough bore pipe. This was achieved by obtaining a series of head loss readings at different flow rates of water flowing through the rough test pipes. A graph of friction coefficient against Reynolds’ number showed that friction coefficient or factor was constant beyond a particular value of Reynolds’ number (about 10³). Therefore once the Reynolds’ number of a rough pipe is known, it is possible to predict the friction factor of that particular pipe and the expected head loss due to water flow through the pipe.

Aim of Experiment

The main objective of this experiment is to determine the relationship between coefficient of friction of the fluid and Reynold’s number for water flowing through a rough pipe. This is attained by taking and recording head loss readings at different flow rates of water flowing through the rough test pipes.  

Brief Introduction/Background

Frictional head loss in a pipe is calculated using equation 4 below

Where = frictional head loss (m), f = friction coefficient of the pipe, L = length of the pipe from one tapping to another (m), u = mean velocity of water flowing through the pipe (m/s), g = acceleration due to gravity (m/s²), and d = internal diameter of the pipe (m). In most cases, λ can be expressed in form of f as λ = 4f.

Reynolds’ number can be calculated using equation 5 below

Where Re = Reynolds’ number, ρ = density (999 kg/m³ at a temperature of 15°), u = mean velocity of water flowing through the pipe (m/s), d = internal diameter of the pipe (m), and μ = molecular velocity (1.15 x 10^-3 Ns/m² at a temperature of 15°C).

Description of Apparatus

The apparatus used in this experiment is Armfield C6-MKII-10 Fluid Friction Apparatus together with Armfield F1-10 Hydraulics Bench. Other devices used are internal vernier caliper and stop watch. The pipes used in this experiment are assumed to have constant internal diameters.

The pipe network was primed with water. Appropriate valves were opened and closed so as to obtain the required water flow through the only roughened test pipe. Readings were taken at different flow rates, with the flow being changed using control valves fitted on the hydraulics bench. Volumetric tank or measuring cylinder was used to measure flow rates. Head loss from one tapping to another was also measured using manometer, sensors or handheld meter. A vernier caliper was used to estimate the test pipe’s nominal internal diameter. The values obtained were then used to estimate the pipe’s roughness factor k/d.

 Data, Results and Graphs

The internal diameter of pipe 7, d, is assumed to be 15.2 mm = 0.0152 m and length of the pipe, p = 1 m.

Table 6 below shows data for pipe 7

Table 6: experimental data for pipe 7

Flow Rate Q (m3/sec)	Measured Head loss (m)	u	Re	f
3.433E-04	0.849505	1.892E+00	2.498E+04	1.803E-02
4.656E-04	1.397388	2.566E+00	3.388E+04	1.613E-02
3.236E-04	0.946864	1.783E+00	2.354E+04	2.263E-02
2.977E-04	0.656696	1.641E+00	2.166E+04	1.854E-02
1.909E-04	0.313076	1.052E+00	1.389E+04	2.150E-02
1.591E-04	0.1899455	8.767E-01	1.158E+04	1.878E-02
1.258E-04	0.0964045	6.930E-01	9.151E+03	1.525E-02
8.055E-05	0.0467705	4.439E-01	5.861E+03	1.804E-02
1.495E-05	0.0047725	8.241E-02	1.088E+03	5.341E-02
3.866E-05	0.0047725	2.130E-01	2.813E+03	7.992E-03
3.514E-05	0.0047725	1.937E-01	2.557E+03	9.670E-03
3.182E-04	0.8065525	1.753E+00	2.315E+04	1.994E-02
1.591E-04	0.1479475	8.767E-01	1.158E+04	1.463E-02
1.193E-04	0.089723	6.575E-01	8.682E+03	1.577E-02
7.342E-05	0.043907	4.046E-01	5.343E+03	2.038E-02
6.818E-05	0.032453	3.757E-01	4.961E+03	1.747E-02
2.450E-05	0.007636	1.350E-01	1.783E+03	3.184E-02
2.580E-05	0.0066815	1.422E-01	1.877E+03	2.512E-02
1.145E-05	0.005727	6.312E-02	8.335E+02	1.092E-01
8.447E-06	0.0047725	4.655E-02	6.147E+02	1.674E-01
4.454E-04	1.765825	2.455E+00	3.241E+04	2.227E-02
4.136E-04	1.536745	2.279E+00	3.010E+04	2.248E-02
3.818E-04	1.326755	2.104E+00	2.778E+04	2.278E-02
3.500E-04	1.116765	1.929E+00	2.547E+04	2.282E-02
3.182E-04	0.925865	1.753E+00	2.315E+04	2.289E-02
2.864E-04	0.754055	1.578E+00	2.084E+04	2.301E-02
2.545E-04	0.5965625	1.403E+00	1.852E+04	2.304E-02
2.227E-04	0.45816	1.227E+00	1.621E+04	2.311E-02
1.909E-04	0.3388475	1.052E+00	1.389E+04	2.327E-02
1.591E-04	0.1384025	8.767E-01	1.158E+04	1.369E-02
1.432E-04	0.1909	7.890E-01	1.042E+04	2.330E-02
1.273E-04	0.15272	7.014E-01	9.261E+03	2.360E-02
1.114E-04	0.1193125	6.137E-01	8.103E+03	2.408E-02
9.545E-05	0.0906775	5.260E-01	6.946E+03	2.491E-02
7.954E-05	0.066815	4.383E-01	5.788E+03	2.643E-02
7.159E-05	0.05727	3.945E-01	5.209E+03	2.797E-02
6.363E-05	0.047725	3.507E-01	4.630E+03	2.949E-02
5.568E-05	0.0429525	3.068E-01	4.052E+03	3.467E-02
4.773E-05	0.028635	2.630E-01	3.473E+03	3.146E-02
2.273E-05	0.002	1.252E-01	1.654E+03	9.690E-03
1.267E-05	0.001	6.980E-02	9.217E+02	1.560E-02
1.933E-06	0.004	1.065E-02	1.407E+02	2.678E+00
3.977E-05	0.01909	2.192E-01	2.894E+03	3.020E-02
3.182E-05	0.01909	1.753E-01	2.315E+03	4.719E-02
2.386E-05	0.01909	1.315E-01	1.736E+03	8.390E-02
1.591E-05	0.0162265	8.767E-02	1.158E+03	1.605E-01
1.432E-05	0.009545	7.890E-02	1.042E+03	1.165E-01
1.273E-05	0.0085905	7.014E-02	9.261E+02	1.327E-01
3.595E-06	0.0047725	1.981E-02	2.616E+02	9.239E-01
3.754E-06	0.0047725	2.069E-02	2.732E+02	8.473E-01
1.273E-04	0.122176	7.014E-01	9.261E+03	1.888E-02
1.909E-04	0.313076	1.052E+00	1.389E+04	2.150E-02
3.471E-04	0.937319	1.913E+00	2.526E+04	1.947E-02
4.088E-04	1.3028925	2.253E+00	2.975E+04	1.951E-02
3.433E-04	0.7550095	1.892E+00	2.498E+04	1.603E-02
3.471E-04	1.032769	1.913E+00	2.526E+04	2.145E-02
5.663E-06	0.005727	3.121E-02	4.121E+02	4.468E-01
3.182E-04	0.8065525	1.753E+00	2.315E+04	1.994E-02
4.136E-04	1.231305	2.279E+00	3.010E+04	1.801E-02
3.341E-06	0.0047725	1.841E-02	2.431E+02	1.070E+00
1.074E-05	0.0047725	5.918E-02	7.814E+02	1.036E-01
1.551E-05	0.0104995	8.548E-02	1.129E+03	1.092E-01
2.335E-05	0.0181355	1.287E-01	1.699E+03	8.325E-02
2.991E-05	0.013363	1.648E-01	2.176E+03	3.739E-02
3.818E-05	0.0047725	2.104E-01	2.778E+03	8.193E-03
5.785E-05	0.047725	3.188E-01	4.209E+03	3.569E-02
7.954E-05	0.03818	4.383E-01	5.788E+03	1.510E-02
1.061E-04	0.07636	5.845E-01	7.717E+03	1.699E-02
6.937E-04	3.312115	3.823E+00	5.048E+04	1.722E-02
3.579E-06	0.0047725	1.973E-02	2.605E+02	9.322E-01
6.395E-06	0.005727	3.524E-02	4.654E+02	3.504E-01
6.250E-04	2.586	3.444E+00	4.548E+04	1.657E-02
4.545E-04	2.139	2.505E+00	3.308E+04	2.591E-02
3.333E-04	1.432	1.837E+00	2.426E+04	3.225E-02
2.778E-04	0.909	1.531E+00	2.021E+04	2.948E-02
2.500E-04	0.589	1.378E+00	1.819E+04	2.358E-02
1.623E-05	0.0181355	8.942E-02	1.181E+03	1.724E-01
1.909E-05	0.0181355	1.052E-01	1.389E+03	1.245E-01
4.009E-05	0.020999	2.209E-01	2.917E+03	3.270E-02
4.104E-05	0.022908	2.262E-01	2.987E+03	3.403E-02
5.154E-05	0.0391345	2.840E-01	3.751E+03	3.686E-02
5.918E-05	0.047725	3.261E-01	4.306E+03	3.410E-02
1.050E-04	0.066815	5.786E-01	7.640E+03	1.517E-02
1.260E-04	0.104995	6.943E-01	9.168E+03	1.655E-02
3.355E-04	0.715875	1.849E+00	2.441E+04	1.592E-02
3.391E-04	0.734965	1.869E+00	2.467E+04	1.600E-02
4.977E-04	1.60356	2.743E+00	3.621E+04	1.620E-02
5.966E-04	2.300345	3.288E+00	4.341E+04	1.618E-02
6.263E-04	3.03531	3.452E+00	4.557E+04	1.936E-02
6.877E-04	3.0544	3.790E+00	5.004E+04	1.616E-02
7.286E-04	3.426655	4.015E+00	5.302E+04	1.615E-02
7.522E-04	4.132985	4.145E+00	5.473E+04	1.828E-02
3.985E-04	1.154945	2.196E+00	2.900E+04	1.820E-02
8.829E-06	0.005727	4.866E-02	6.425E+02	1.838E-01
1.814E-05	0.0181355	9.994E-02	1.320E+03	1.380E-01
2.864E-05	0.013363	1.578E-01	2.084E+03	4.078E-02
3.245E-05	0.011454	1.788E-01	2.362E+03	2.722E-02
3.484E-05	0.011454	1.920E-01	2.535E+03	2.361E-02
7.018E-05	0.028635	3.868E-01	5.107E+03	1.455E-02
9.312E-05	0.05727	5.132E-01	6.776E+03	1.653E-02
2.357E-04	0.353165	1.299E+00	1.715E+04	1.591E-02
3.767E-04	0.906775	2.076E+00	2.741E+04	1.599E-02
5.492E-04	2.08081	3.027E+00	3.996E+04	1.726E-02
4.167E-05	0.033	2.296E-01	3.032E+03	4.757E-02

A graph of friction coefficient of the pipe against Reynolds’ number is as shown in Figure 8 below

Generally, head loss increases with increase in flow rate through a roughened pipe. This is simply because when flow rate increases, it means that volume of water through the pipe and the mean velocity of the water have also increased (Furuichi, et al., 2015). This increase in velocity and volume of water through the pipe causes an increase in contact area between water and internal surface of the pipe thus increasing frictional coefficient (Nazemi, et al., 2011). Figure 8 above shows a graph of pipe friction coefficient against Reynolds’ number. The graph shows that pipe friction coefficient starts by decreasing with increasing Reynolds’ number before becoming constant even with increasing Reynold’s number past a particular value of Reynolds’ number (a higher Reynolds’ number) (Masuyama & Hatakeyama, 2009). This basically means that at high Reynolds’ number, friction coefficient of the pipe becomes constant (Furuichi, et al., 2009). At higher Reynolds’ number, the flow becomes more stable thus reducing the bombardment of water on internal pipe surface (Ahsan, 2014).     

Conclusion

Once Reynolds’ number of water flowing through a pipe is known, friction coefficient of the fluid can be obtained from Moody’s diagram. A graph of friction coefficient against Reynolds’ number obtained in this experiment showed that friction coefficient or factor remains constant beyond a particular value of Reynolds’ number (about 10³ in this experiment). Therefore once the Reynolds’ number of a rough pipe is known, it is possible to predict the friction factor of that particular pipe and the expected head loss due to water flow through the pipe.

References

Ahsan, M., 2014. Numerical analysis of friction factor for a fully developed turbulent flow using k–ε turbulence model with enhanced wall treatment. Beni-Suef University Journal of Basic and Applied Sciences, 3(4), pp. 269-277.

Burger, J., Haldenwang, R. & Alderman, N., 2010. Friction factor-Reynolds number relationship for laminar flow of non-Newtonian fluids in open channels of different cross-sectional shapes. Chemical Engineering Science, 65(11), pp. 3549-3556.

Cerbus, R., Liu, C., Gioia, G. & Chakraborty, P., 2018. Laws of Resistance in Transition Pipe Flows. Physical Review Letters, 120(5-2).

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