The Loss Coefficient

Looking at the Bernoulli equation, we see that the friction loss (ef) can be made dimensionless by dividing it by the kinetic energy per unit mass of fluid. The result is the dimensionless loss coefficient, K  

A loss coefficient can be defined for any element that offers resistance to flow (i.e., in which energy is dissipated), such as a length of conduit, a valve, a pipe fitting, a contraction, or an expansion. The total friction loss can thus be expressed in terms of the sum of the losses in each element, i.e., ef = JT K-nVf/7). This will be discussed further in Chapter 6. 

Although pV2/2 represents kinetic energy per unit volume, pV2 is also the flux of momentum carried by the fluid along the conduit. The latter interpretation is more logical in Eq. (5-50), because rw is also a flux of momentum from the fluid to the tube wall. However, the conventional definition includes the (arbitrary) factor i. Other definitions of the pipe friction factor are in use that are some multiple of the Fanning friction factor. For example, the Darcy friction factor, which is equal to 4/, is used frequently by mechanical and civil engineers. Thus, it is important to know which definition is implied when data for friction factors are used. 

Because the friction loss and wall stress are related by Eq. (5-47), the loss coefficient for pipe flow is related to the pipe Fanning friction factor as follows  

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Miller Internal Flow Systems, 2d ed.. Chap. 13, BHRA, Cranfield, 1990) gives the most complete information on losses in bends and curved pipes. For turbulent flow in circular cross-seclion bends of constant area, as shown in Fig. 6-14 7, a more accurate estimate of the loss coefficient K than that given in Table 6-4 is... 

Now it appears that the value of may be estimated by using the loss coefficient K determined at choking provided K is not too small. This is unlikely since in most valves effective flow control occurs at very small throat area when the valve is in the 10-30% open range. The loss coefficient is determined from the pressure loss across the valve and the velocity in the upstream pipe at choking. 

Because most applications for micro-channel heat sinks deal with liquids, most of the former studies were focused on micro-channel laminar flows. Several investigators obtained friction factors that were greater than those predicted by the standard theory for conventional size channels, and, as the diameter of the channels decreased, the deviation of the friction factor measurements from theory increased. The early transition to turbulence was also reported. These observations may have been due to the fact that the entrance effects were not appropriately accounted for. Losses from change in tube diameter, bends and tees must be determined and must be considered for any piping between the channel plenums and the pressure transducers. It is necessary to account for the loss coefficients associated with singlephase flow in micro-channels, which are comparable to those for large channels with the same area ratio. 

Table 13.4 gives some typical values of the loss coefficient for various fittings9. It should be noted that values for loss coefficient will vary for the same fitting, but from different manufacturers, as a result of differences in geometry. Table 13.5 gives head losses for sudden contractions, sudden expansions and orifice plates. Note that the relationship for orifice plates in Table 13.5 relates to the overall pressure drop and not the pressure drop between the pressure tappings used to determine the flowrate. 

Example 5-6 Friction Loss in a Sudden Expansion. Figure 5-7 shows the flow in a sudden expansion from a small conduit to a larger one. We assume that the conditions upstream of the expansion (point 1) are known, as well as the areas A and A2. We desire to find the velocity and pressure downstream of the expansion (V2 and P2) and the loss coefficient, Kt. As before, V2 is determined from the mass balance (continuity equation) applied to the system (the fluid in the shaded area). Assuming constant density,... 

The loss coefficient is seen to be a function only of the geometry of the system (note that the assumption of plug flow implies that the flow is highly turbulent). For most systems (i.e., flow in valves, fittings, etc.), the loss coefficient cannot be determined accurately from simple theoretical concepts (as in this case) but must be determined empirically. For example, the friction loss in a sudden contraction cannot be calculated by this simple method due to the occurrence of the vena contracta just downstream of the contraction (see Table 7-5 in Chapter 7 and the discussion in Section IV of Chapter 10). For a sharp 90° contraction, the contraction loss coefficient is given by... 

Methods for evaluating the loss coefficient K( will be discussed in Chapter 6. 

You have probably noticed that when you turn on the garden hose it will whip about uncontrollably if it is not restrained. This is because of the unbalanced forces developed by the change of momentum in the tube. If a 1/2 in. ID hose carries water at a rate of 50 gpm, and the open end of the hose is bent at an angle of 30° to the rest of the hose, calculate the components of the force (magnitude and direction) exerted by the water on the bend in the hose. Assume that the loss coefficient in the hose is 0.25. 

Repeat Problem 46, for the case in which a nozzle is attached to the end of the hose and the water exits the nozzle through a 1/4 in. opening. The loss coefficient for the nozzle is 0.3 based on the velocity through the nozzle. 

You are watering your garden with a hose that has a 3/4 in. ID, and the water is flowing at a rate of 10 gpm. A nozzle attached to the end of the hose has an ID of 1 /4 in. The loss coefficient for the nozzle is 20 based on the velocity in the hose. Determine the force (magnitude and direction) that you must apply to the nozzle in order to deflect the free end of the hose (nozzle) by an angle of 30° relative to the straight hose. 

A 4 in. ID fire hose discharges water at a rate of 1500 gpm through a nozzle that has a 2 in. ID exit. The nozzle is conical and converges through a total included angle of 30°. What is the total force transmitted to the bolts in the flange where the nozzle is attached to the hose Assume the loss coefficient in the nozzle is 3.0 based on the velocity in the hose. 

A 90° horizontal reducing bend has an inlet diameter of 4 in. and an outlet diameter of 2 in. If water enters the bend at a pressure of 40 psig and a flow rate of 500 gpm, calculate the force (net magnitude and direction) exerted on the supports that hold the bend in place. The loss coefficient for the bend may be assumed to be 0.75 based on the highest velocity in the bend. 

A fireman is holding the nozzle of a fire hose that he is using to put out a fire. The hose is 3 in. in diameter, and the nozzle is 1 in. in diameter. The water flow rate is 200 gpm, and the loss coefficient for the nozzle is 0.25 (based on the exit velocity). How much force must the fireman use to restrain the nozzle Must he push or pull on the nozzle to apply the force What is the pressure at the end of the hose where the water enters the nozzle ... 

Water flows through a 45° expansion pipe bend at a rate of 200 gpm, exiting into the atmosphere. The inlet to the bend is 2 in. ID, the exit is 3 in. ID, and the loss coefficient for the bend is 0.3 based on the inlet velocity. Calculate the force (magnitude and direction) exerted by the fluid on the bend relative to the direction of the entering stream. 

A relief valve is mounted on the top of a large vessel containing hot water. The inlet diameter to the valve is 4 in., and the outlet diameter is 6 in. The valve is set to open when the pressure in the vessel reaches 100 psig, which happens when the water is at 200° F. The liquid flows through the open valve and exits to the atmosphere on the side of the valve, 90° from the entering direction. The loss coefficient for the valve has a value of 5, based on the exit velocity from the valve. 

A relief valve is installed on the bottom of a pressure vessel. The entrance to the valve is 4.5 in. diameter, and the exit (which discharges in the horizontal direction, 90° from the entrance) is 5 in. in diameter. The loss coefficient for the valve is 4.5 based on the inlet velocity. The fluid in the tank is a liquid with a density of 0.8 g/cm3. If the valve opens when the pressure at the valve reaches 150 psig, determine ... 

An emergency relief valve is installed on a reactor to relieve excess pressure in case of a runaway reaction. The lines upstream and downstream of the valve are 6 in. sch 40 pipe. The valve is designed to open when the tank pressure reaches 100 psig, and the vent exhausts to the atmosphere at 90° to the direction entering the valve. The fluid can be assumed to be incompressible, with an SG of 0.95, a viscosity of 3.5 cP, and a specific heat of 0.5 Btu/(lbm °F). If the sum of the loss coefficients for the valve and the vent line is 6.5, determine ... 

The 2-K method was published by Hooper (1981, 1988) and is based on experimental data in a variety of valves and fittings, over a wide range of Reynolds numbers. The effect of both the Reynolds number and scale (fitting size) is reflected in the expression for the loss coefficient ... 

A note is in order relative to the exit loss coefficient, which is listed in Table 7-5 as equal to 1.0. Actually, if the fluid exits the pipe into unconfined space, the loss coefficient is zero, because the velocity of a fluid exiting the pipe (in a free jet) is the same as that of the fluid inside the pipe (and the... 

The flow rate can be readily calculated if the loss coefficients can be determined. The procedure involves an iteration, starting with estimated values for the loss coefficients. These are used in Eq. (7-47) to find Q, which is used to calculate the Reynolds number(s), which are then used to determine revised values for the A) s, as follows. 

A commercial steel (e = 0.0018 in.) pipeline is 1 in. sch 40 diameter, 50 ft long, and includes one globe valve. If the pressure drop across the entire line is 22.1 psi when it is carrying water at a rate of 65 gpm, what is the loss coefficient for the globe valve The friction factor for the pipe is given by the equation... 

One way to achieve the desired flow rate of 275 gpm would obviously be to close down on the valve until this value is achieved. This is equivalent to increasing the resistance (i.e., the loss coefficient) for the system, which will shift the system curve upward until it intersects the 7 in. impeller curve at the desired flow rate of 275 gpm. The pump will still provide 250 ft of head, but about 30 ft of this head is lost (dissipated) due to the additional... 

Just as for isothermal flow, this is an implicit expression for the choke pressure (P ) as a function of the upstream pressure (Pi), the loss coefficients (J] Kf), and the isentropic exponent (7c), which is most easily solved by iteration. It is very important to realize that once the pressure at the end of the pipe falls to P and choked flow occurs, all of the conditions within the pipe (G = G, P2 = P, etc.) will remain the same regardless of how low the pressure outside the end of the pipe falls. The pressure drop within the pipe (which determines the flow rate) is always Pt — P when the flow is choked. 

C0 is obviously a function of ft and the loss coefficient K (which depends on... 

If the loss coefficient is based upon the velocity through the orifice (VD) instead of the pipe velocity, the /l4 term in the denominator of Eq. (10-22) does not appear ... 

This equation defines the flow coefficient, Cv. Here, SG is the fluid specific gravity (relative to water), pw is the density of water, and hv is the head loss across the valve. The last form of Eq. (10-29) applies only for units of Q in gpm and hv in ft. Although Eq. (10-29) is similar to the flow equation for flow meters, the flow coefficient Cv is not dimensionless, as are the flow meter discharge coefficient and the loss coefficient (Af), but has dimensions of [L3][L/M]1/2. The value of Cv is thus different for each valve and also varies with the valve opening (or stem travel) for a given valve. Values for the valve Cv are determined by the manufacturer from measurements on each valve type. Because they are not dimensionless, the values will depend upon the specific units used for the quantities in Eq. (10-29). More specifically, the normal engineering (inconsistent) units of Cv are gpm/ (psi)1/2. [If the fluid density were included in Eq. (10-29) instead of SG, the dimensions of Cv would be L2, which follows from the inclusion of the effective valve flow area in the definition of Cv]. The reference fluid for the density is water for liquids and air for gases. 

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