The smooth flow in pipes can be divided into laminar and turbulent flow using the critical velocity. The flow pattern is determined by the diameter of the pipe, flow rate, density and viscosity (Assefa and Kaushal). Thus a set of dimensionless groups can be determined using the following variables
Where
is the diameter of the pipe
is the viscosity of the fluid
is the average velocity of the fluid
The Reynolds number is used in the classification of the dimensionless groups. The laminar flow will occur when the Reynolds number is below 2100. In case of a stable supply state, the Re will be a few thousand. In the case of general pipes, the Re will be greater than 4000 (Marusic, Joseph and Mahesh). This is called the transition region, i.e., between 2100 and 4000. In this region, the flow can either be turbulent or laminar based on the entry state.
Friction Factor
The friction factor can be illustrated as
Where
is the pressure drop
is the length of the pipe
The friction factor is a function of Reynolds number expressed in Newtonian fluid flow through a smooth pipe. The flow occurs between ends of the pipe; thus the variables associated with the system will be defined via the dimensional analysis (Garcı´a, Garcı´a, and Garcı´a). The Friction Factor-Reynolds number relationship data will provide the relation between the actual pressure difference and the velocity of the fluid. The above relationship will be determined via experimental analysis. The plot of Friction-Reynolds produces two regions. In the plot laminar flow is Re<2100, with a slope -1 in the log scale. The turbulent flow will be recorded when the Re is greater than 4000. In the turbulent region, there is no match between the friction factor and the Reynolds number (Garcı´a, Garcı´a, and Garcı´a). In this region there the flow pattern is extremely chaotic. Experimental conditions determine the transition from laminar to turbulent flow. In this region, it is typically difficult to plot f as a function of the Reynolds number. The fully developed turbulent flow is found between Re>4000. Thus, the relationship between friction factor and the Reynolds number can be expressed using the equation below.
Thus from the definition f and Re we can rewrite the above equation as
The above relation is referred to as the Hagen-Poiseuille equation. In case of the turbulent flow, the relation will be represented as
The equation can be rearranged with respect as Q as illustrated below
The above relation is known as the Blasius equation. The equation does not have a theoretical background, and it is often applied to the flow in the region of high Reynolds number (Marusic, Joseph and Mahesh). The following relationship can be utilized as
The above relation is known as von Karman-Nikuradse equation. The equation is an implicit friction factor. The term in the relation makes velocity irrelevant.
End Effects
The initial evaluation was for a fully developed flow. At the entrance or the exit, the flow is not fully developed. For the f-relation with a known entrance length can be expressed as
The exit length is negligible when compared in size with a pipe diameter becoming meaningless when
Power and dissipation
The power consumed can be calculated from the work done
The law of conservation of energy states that work done by the pressure difference should increase the energy of the system will be dissipated as heat. Additionally, the work dissipated per unit volume of the fluid can be calculated using the power consumption per unit volume (Assefa and Kaushal). Considering the laminar flow, the dissipated energy will be calculated as
Re-arranging the to make the subject of the equation from the definition of the friction factor.
Inserting the into the dissipation equation
Commercial Pipe
The considerations above have been for a smooth pipe. The commercial pipes incorporate a roughness k that should be considered for the inner surfaces (Garcı´a, Garcı´a and Garcı´a). For laminar flow the friction factor is independent of roughness k/D and has the same value. The roughness is traditionally evident in the turbulent flow thus the Colebrook formula can be utilized to incorporate roughness factor.
For then the equation for the commercial pipe can be expressed as
Experimental Procedure
Operation mode
The viscometer was leveled using the two levelling screw located at the base. The adjustment was made so that the bubble levels on the top of the DV-II+Pro which can be centered within the circle.
The AC power switch at the rear of the DV-II+Pro was set at OFF. Then the power code was connected to the socket located that the back panel of the instrument and plugged into the appropriate AC line.
The power switch was turned ON and allowed to warm the device for 10 minutes before it was performed to autozero. The viscometer was autozeroed before any reading was taken. This action was performed every time the device was switched ON.
The spindle was removed and then a key pressed then the DV-II+Pro will commence autozeroing. This will be shown by the screen flashing autozeroing.
The spindle was attached to the viscometer by screwing them onto the coupling nut on the lower shaft. The spindle has a left-hand thread. The lower shaft of the coupling should be secured and slightly lifted with one hand when screwing the spindle to the left. The matching surface on the lower shaft and face of the spindle nut should be smooth and cleaned to prevent eccentric rotation of the spindle.
The DV-II+Pro must have a spindle Entry Code number to calculate the viscosity, shear stress values and shear rate. The DV-II+Pro memory contains parameters for common Brookfield spindles that includes two-digit entry code for each spindle and custom spindles. In the case of the UL-Adapter the spindle Entry code will be 00.
The selection of a viscometer speeds the UP or DOWN arrow was pressed. This will cause the area to the right of RPM to display the current selected speed.
The motor was turned on
The time for the indicated reading was allowed to stabilize. The time required for stabilization will be dependent on the speed the viscometer was running and the characteristic of the sample fluid. For maximum accuracy any reading below 10% should be avoided.
The percentage torque was recorded and viscosity
The device was turned off.
Procedure
The flow line of ¼ in or 3/8 was selected by a 3-way valve that was located under the reservoir. The flow rate in the experiment can be controlled using a needle valve and a pump (Marusic , Joseph and Mahesh). The experimental procedure for data collection is as follows.
The reservoir was filled
The flow line was changed using a 3-way-valve to the ¼ inch tube
The difference pressure transmitter, flowmeter, pump, COM and DAQ was turned on.
The needle valve was opened once the reservoir was full. The needle valve connected to the pressure transmitter was opened to fill the tube with the fluid.
The fluid was allowed to flow at a constant flow rate. Once the fluid stabilized the and Q were recorded
The above procedures were repeated for the 3/8-inch tube.
The measurement temperature and the viscosity of the water was maintained similar to the temperature of water.
The fluid was drained and the tube dried by blowing air through it. The steps above were repeated for ethylene glycol.
The viscosity of ethylene glycol was measured
Results
volumatric flow rate [mL/min]
volumatric flow rate [m^3/s]
flow velocity [m/s]
Re [-]
f [f]
100
1.667E-06
0.1071615
476.8688426
0.0335522
200
3.333E-06
0.2143231
953.7376851
0.0167761
300
0.000005
0.3214846
1430.606528
0.0111841
400
6.667E-06
0.4286462
1907.47537
0.0083881
Table 1: Results for the laminar flow in water
volumatric flow rate [mL/min]
volumatric flow rate [m^3/s]
flow velocity [m/s]
Re [-]
f [f]
1500
0.000025
0.5467655
4171.821132
0.000983
2000
3.333E-05
0.7290207
5562.428177
0.0009148
2500
4.167E-05
0.9112759
6953.035221
0.0008651
3000
0.00005
1.0935311
8343.642265
0.0008266
Table 2: Results for the turbulent flow in water
volumatric flow rate [mL/min]
volumatric flow rate [m^3/s]
flow velocity [m/s]
Re [-]
f [f]
100
1.667E-06
0.036451
19.27847778
0.0037702
200
3.333E-06
0.0729021
38.55695556
0.0031703
300
0.000005
0.1093531
57.83543334
0.0028647
400
6.667E-06
0.1458041
77.11391112
0.0026659
Table 3: Results for the laminar flow in Ethylene glycol
Figure 1: The graph of friction as a factor of Reynolds number for laminar flow in water
Figure 2: The graph of friction as a factor of Reynolds number for turbulent flow in water
Figure 3: The graph of friction as a factor of Reynolds number for turbulent flow in Ethylene Glycol
Discussion
The results above show the charts are a representation of the Moody chart where the data below 2000 was close to the function f=64/R. The slight deviation in the graph might have been recorded by the errors that occurred in the experiment (Garcı´a, Garcı´a, and Garcı´a). The water flow in the pipes was unstable when in transition and the value of the Re was between 2100 and 4000. In this case, the friction factor kept on fluctuating. The experiment was conducted on minor errors, and the transition was observed for higher values for the laminar and the turbulent flows. The ε/D was estimated to be between 0.004 and 0.001. The values were obtained from the estimated from the graph. The volumetric flow rate Q and friction related in a similar fashion. The water used in the experiment was not subjected to the ideal conditions. The water might have not been pure with dissolved chemicals and other substances. The presence of these substances affected the viscosity of water considerably (Marusic, Joseph and Mahesh). The pressure of the water in the pipe is expected to fluctuate. There might be some instances of human errors in the experiment. The errors discussed above would affect the results considerably.
Conclusion
The experiment was successful in analyzing fluid as a function of the Reynolds number. The theoretical relationship was evaluated to provide a graph that resemble the Moody Chart. The transition occurred in the region between Re 2100 and 4000 which is in agreement with the theory.
Works Cited
Assefa, K M and D R Kaushal. "A comparative study of friction factor correlations for high concentrate slurry flow in smooth pipes." Journal of Hydrology and Hydromechanics 63.1 (2015): 13-20. Print.
Garcı´a, F et al. "Friction factor improved correlations for laminar and turbulent gas-liquid flow in horizontal pipelines." International Journal of Multiphase Flow 33 (2007): 1320-1336. Print.
Marusic, Ivan, D D Joseph, and Krishnan Mahesh. "Laminar and turbulent comparisons for channel flow and flow control." Journal of Fluid Mechanics 570 (2007): 467-477. Print.
