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MAJOR AND MINOR LOSSES OF INTERNAL FLOW

C. J. Coupland

ABSTRACT. The loss coefficients of an elbow, a tee junction, and a gate valve fully open, two-thirds closed, and five-sixths closed are determined and major and minor losses are compared. Head loss can be broken down into major losses and minor losses. Major losses are head losses due to friction and are related by the dimensionless friction factor. Minor losses are head losses due to flow through fittings and are related by loss coefficient. Minor losses are a larger component in total head loss for systems with many fitting and few long straight sections of pipes. The values for loss coefficient are calculated for each fitting tested and the friction factor is calculated for the pipe at two lengths. These values are compared to the theoretical values found using the Colebrook equation and the loss coefficient estimation table. Comparing the loss coefficients calculated from testing and the theoretical values it is clear the tables are an inaccurate estimate. The Colebrook equation is found to give a close estimate of friction factor although it too varies from the calculated value.

istribution of fluid flow is common in municipal water networks, heating and cooling systems, and irrigation systems. Network pipe systems are composed of sections of pipes and fittings that direct the flow. The flow through various fittings, valves, bends, elbows, and tees alters the flow

pattern, creates separation, and turbulence in the system. The network components create irreversible head losses know as minor losses, while the frictional head losses in straight sections are referred to as major losses (Haghighi et al. 1988). Although they are termed “minor” the cumulative effect of several components is often substantial. The term minor loss is a misnomer as the effect usually is major (Gerrish et al. 1995). In order to model a system with the highest degree of accuracy it is necessary to account for the pressure drop due to both major and minor losses. Head loss is necessary to determine the pumping power requirement of a system and is equivalent to the additional height that the fluid needs to raised by a pump in order to overcome the major and minor losses in the pipe (Çengel and Cimbala, 2010).

The head loss is measured using a manometer. A manometer is a simple, accurate, and cost effective pressure measurement system that consists of a glass or plastic U-tube that contains air or an immiscible fluid. The pressure differences in the pipe are reflected in the U-tube and the head loss can be measured.

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Minor losses are usually expressed in terms of loss coefficient, and are defined as:

𝐾! =  ℎ!

𝑉!2𝑔

where, KL = the loss coefficient hL = the additional head loss caused by the component V = average velocity of the flow g = the acceleration due to gravity

The loss coefficient value is dependent on the geometry of the component, and flow characteristics. The values can be estimated using a table of values but this method results in a high degree of uncertainty. Manufacturer testing data should be consulted rather than relying on the represenitive data (Rochester et al. 1989).

The magnitude of the major losses depends on the friction factor. The friction factor is a function of the dimensions of the pipe, the material the pipe is made out of, and the Reynolds number (Irwin and Tsang, 1971). A fluid flow with a Reynolds number greater than 4000 is considered turbulent and disordered (Allen, 1996). The inertial forces are large relative to the viscous forces, and cannot prevent random and rapid fluctuations in the fluid. As a result turbulent flows are associated with much higher values for friction (Çengel and Cimbala, 2010). The primary objectives of this work were to determine the loss coefficient values for various fittings, to compare the major and minor losses, and become familiar with the principals of internal flow. The volumetric flow rates used in testing will be calculated using a venturi flow meter that has previously been calibrated. Minor head loss was measured across a gate valve, tee junction, and elbow, while major head loss was measured across a pipe section for each test. The loss factor and friction factor found from testing will be compared to the represenitive loss coefficient values and calculated values from the Colebrook equation. MATERIALS AND METHODS

A Technovate fluid circuit system (fig. 1) was used for our testing. The flow in the system was controlled by opening control valve 52. The water was directed through the fluid system and to the pipe section. The water left the pipe section and passed through a gate valve, tee junction, an elbow, and a venturi flow meter before returning to the tank. The venturi flow meter was attached to a digital manometer and has been calibrated in previous experiments, yielding an expression for the volumetric flow rate as a function of head loss. The minor losses were tested across the gate valve, the tee junction and the elbow by attaching a manometer directly before and after the component to measure the pressure drop. Simultaneously the pressure drop in the pipe section was measured.

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In our testing two manometers were used. In the first set of data the pressure drop over a three-foot section of pipe and a gate valve (fig. 2) fully open, two-thirds closed and five-sixths closed were measured. The next set measured the pressure drop over a tee junction and three-foot section of pipe. The last data measurements were for an elbow and a three-foot as well as five-foot section of pipe. The section of copper pipe we tested had a diameter of 1”.

RESULTS AND DISCUSSION

Analysis of Data Collected For each test the change in height of the manometer attached to the pipe section and pipe fitting are recorded. Across the venturi flow meter the digital manometer measured the head loss and from previous calibration tests flow rate was found using the equation:

𝑄 = 97.689(∆𝑃)!.!"# where, 𝑄 = the volumetric flow rate in gallons per hour ∆𝑃 = the pressure drop in inches of water This equation is solved specifically for the venturi flow meter we are testing with and is used to find volumetric flow rate for each set of data. The data that we recorded in testing along with the flow rates are displayed below (table 1).

Figure1. Technovate fluid circuit system. Image obtained from the University of Portland.

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Figure 2. A gate valve shown about one-third closed. Image obtained from HiFly Valve Co.

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In terms of equivalent fluid column height the major head loss can be expressed by the equation:

ℎ!,!"#$% = 𝑓𝐿𝐷 𝑉!"#

! where, hL = head loss across the pipe section

Vavg = average velocity f = the friction factor

L = the length of the pipe D = the diameter of the pipe The minor losses due to the components of the fluid flow can be expressed by the equation:

ℎ!,!"#$% = 𝐾!𝑉!"#!

2𝑔

where, hL = head loss across the components KL = the loss coefficient Vavg = average velocity

g = the acceleration due to gravity The values for head loss above in equations 3 & 4 are specific to each individual length of pipe or fitting. Therefore the manometer used in measurement had to be placed directly at the inlet and outlet of the elbow, tee junction, and gate valve.

Table 1. Manometer head differential through gate valve, tee junction, elbow, venturi, and pipe section.

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Analysis of Calculated Loss Coefficient and Friction factor

In the pipe system the volumetric flow rate at any point along the pipes was constant and found using equation 2. The volumetric flow rate can be used to calculate average velocity of the flow with the equation:

𝑉!"# =  𝑄𝐴!

where, Vavg = average velocity of the fluid flow  𝑄 = volumetric flow rate Ac = cross sectional area The average velocity is the average speed of the flow through the cross-section of the pipe. Friction factor for the pipe section and loss coefficient for each fitting can be calculated using equations 3 & 4 with the values of average velocity. The values calculated for friction factor and loss coefficient are displayed below (table 2). The value for loss coefficient in the globe valve increases exponentially from an almost negligible K = 0.11, when fully open, to K = 13.26, for five-sixths closed. The testing done on the elbow over the two lengths of pipe at a constant flow rate shows that the major losses increase with length even though the friction factor decreased. The loss coefficient stayed the same in the elbow between the two tests since the flow rate and velocity did not change. For the three-foot lengths of the gate valve fully open, the tee junction and the elbow the flow rates were nearly the same as well as the friction factor. The decrease in velocity of the flow when the gate valve was semi-closed led to a decrease in major loss and increase in the friction factor.

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Table 2. Flow rate, velocity, and calculated loss coefficient and friction factor for gate valve, tee junction and elbow.

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Analysis of Theoretical Loss Coefficient and Friction factor Since there is no equation for calculating the loss coefficient it can be estimated by using a table that contains representative values based on the system geometry. The table of estimated values is not the most accurate way to estimate loss coefficients since values can vary wildly between two similar components. Manufacturer provided testing data specific to the each component is more accurate in predicting the loss coefficient but is not available for the Technovate system. Using the tables found in Fluid Mechanics: Fundamentals and Applications the estimated loss coefficients are found (Çengel and Cimbala, 2010).

Unlike the loss coefficient the friction factor can be calculated using the Colebrook equation. The Reynolds number must first be calculated for each data point by the equation:

𝑅𝑒 =𝜌𝑉!"#𝐷𝜇

where, 𝜌 = density of the fluid Vavg = Average velocity of the fluid D = Hydraulic diameter μ = dynamic viscosity of the fluid

In our analysis we assume the value for density to be 1000 kg/m3 and the value for dynamic viscosity to be 1.002 X 10-3 kg/m s. The friction factor is dependent on the Reynolds number and the pipes relative roughness, ε/D, which is the ratio of the mean height of the roughness to the pipe diameter. Using a table of calculated values, the roughness is found to be 0.0015 mm, or 1.5 X 10-6 m. For turbulent flow the friction factor is found by using the Colebrook equation:

1𝑓= −2.0 log

𝜀𝐷

3.7 +2.51𝑅𝑒 𝑓

where, f = the Darcy-Weisbach friction factor 𝜀

𝐷 = relative roughness of the pipe Re = Reynolds Number An iterative solution is needed to solve the Colebrook equation. Using a goal seek function in excel the friction factor was calculated for each data set. The calculated values found from our testing are shown below along with the theoretical values for friction factor and loss coefficient (table 3). The theoretical values for loss coefficient of a gate valve needed to be interpolated. The theoretical estimates for the threaded bend (K = 0.9) and tee junction in line flow (K = 0.9) both gave values that were fairly close to the calculated values. The friction factor in the pipe for the gate valve tests decreased between fully open to five-sixths closed in both the calculated and estimated value.

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The Colebrook equation is the most accurate equation for estimating the friction factor and is considered accurate to the ±15 percent. The loss coefficient table gives a value that can vary from the actual values. This is evident in the values estimated and the calculated for loss coefficient in our testing. The tables proved to be a rough estimate for the tee junction and elbow. The estimated loss coefficient for the gate valve was not as close as the differed considerably. The total head loss in a piping system is the summation of the minor head losses and the major head losses:

ℎ! = ℎ!,!"#$% + ℎ!,!"#$%

ℎ! = 𝑓𝐿𝐷 𝑉!"#

! + 𝐾!𝑉!"#!

2𝑔

In order to calculate the head loss to the highest degree of accuracy the minor losses must not be ignored. If the system consists of very long straight pipe sections and few fittings in between the minor losses do not create as much head loss compared to the major losses. For a system like the Technovate fluid circuit with many fittings and relatively short straight portions of pipe the total head loss is mainly due to minor losses. Accuracy is desirable in prediction of pressure heads for reducing pumping costs and lowering material cost in development of the system.

Table 3. Theoretical and calculated values for the friction factor and loss coefficient for a gate valve, tee junction, and elbow.  

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SUMMARY AND CONCLUSIONS Minor losses are a result of components such as valves, tee junctions, and elbows interrupting the flow and creating separation and turbulence. These losses along with major losses, or frictional losses, make up the total head loss in a system. In a large system composed of long straight sections of pipes with few fittings in between, minor losses account for a small percent of the head loss. On the other hand a system such as the Technovate system used in testing, minor losses accounted for most of the head loss.

Flow through valves and fittings is very complex and theoretical analysis is not possible. The loss coefficient is much harder to estimate than the friction factor since there is no equation that can be solved to get an approximation. The general assumptions proved to be inexact estimates of the loss coefficient since the values. Testing is the most reliable source for estimates on loss coefficients. From our tests at a volumetric flow rate of about 630 gallons per hour we calculated the loss coefficients to be K = 0.65 for the tee junction, K = 0.4 for the elbow, and K = 0.1 for the open gate valve. The loss coefficient increased to K = 3 for the valve two-thirds closed, and increased to K = 13.25 for the valve five-sixth closed. The effect of minor losses on the total head loss in the system might be negligible for an open valve, K = 0.1, but accounts for most of the total head loss for a partially closed valve, K = 13.25.

In the partially closed gate valve the head loss increased by about a factor of 4 from two-thirds closed to five-sixths closed. The reason for this is because of the fluid is hitting the blunt face of the gate valve and separates while navigating the sharp edge of the valve. A large turbulent region is created in the wake of the valve. The estimates for loss coefficient provided in the tables are closer to the measured values for the elbow and the tee branch then the valve. Since the loss coefficient for valves depends highly on the design, the estimation proved to vary wildly from the calculated value.

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REFERENCES

Allen, R. G. 1996. Relating the Hazen-Williams and Darcy-Weisbach Friction Loss Equations for Pressurized Irrigation. Logan, U.T.: American Society of Agricultural Engineers.

Çengel, Y. A., and J. M. Cimbala. 2010. Fluid Mechanics: Fundamentals and Applications. 2nd ed. New York, N.Y.: McGraw-Hill.

Gerrish, P. J., W. H. Shayya, and V. F. Bralts. 1995. An Improved Method for Incorporating Pipe Components into the Analysis of Hydraulic Networks. Lansing, M.I.: American Society of Agricultural Engineers.

Haghighi, K., V. F. Bralts, and L. J. Sergerlind. 1988. Finite Element Formulation of Tee and Bend Components in Hydraulic Pipe Network Analysis. Lafayette, I.N.: American Society of Agricultural Engineers.

Irwin, R. W., and G. Tsang. 1971. Hydraulic Roughness of Corrugated Plastic Tubing. Guelph, Canada.: American Society of Agricultural Engineers.

Rochester, E. W., C. A. Flood, and S. G. Hackwell. 1989. Pressure Losses from Hose Coiling of Hard-hose Travelers. Auburn, A.L.: American Society of Agricultural Engineers.