Fluid Mechanics 8

As noted previously, turbulent flow can be a very complex, difficult topic—one that as yet has defied a rigorous theoretical treatment. Thus, most turbulent pipe flow analyses are based on experimental data and semi-empirical formulas. These data are expressed conveniently in dimensionless form. 8.4 Dimensional Analysis of Pipe Flow Chaos theory may eventually provide a deeper understanding of turbulence. JWCL068_ch08_383-460.qxd 9/23/08 10:52 AM Page 409 It is often necessary to determine the head loss, , that occurs in a pipe flow so that the energy equation, Eq. 5.84, can be used in the analysis of pipe flow problems. As shown in Fig. 8.1, a typical pipe system usually consists of various lengths of straight pipe interspersed with various types of components (valves, elbows, etc.). The overall head loss for the pipe system consists of the head loss due to viscous effects in the straight pipes, termed the major loss and denoted , and the head loss in the various pipe components, termed the minor loss and denoted . That is, The head loss designations of "major" and "minor" do not necessarily reflect the relative importance of each type of loss. For a pipe system that contains many components and a relatively short length of pipe, the minor loss may actually be larger than the major loss. 8.4.1 Major Losses A dimensional analysis treatment of pipe flow provides the most convenient base from which to consider turbulent, fully developed pipe flow. An introduction to this topic was given in Section 8.3. As is discussed in Sections 8.2.1 and 8.2.4, the pressure drop and head loss in a pipe are dependent on the wall shear stress, between the fluid and pipe surface. A fundamental difference between laminar and turbulent flow is that the shear stress for turbulent flow is a function of the density of the fluid, For laminar flow, the shear stress is independent of the density, leaving the viscosity, as the only important fluid property. Thus, as indicated by the figure in the margin, the pressure drop, for steady, incompressible turbulent flow in a horizontal round pipe of diameter D can be written in functional form as (8.32) where V is the average velocity, is the pipe length, and is a measure of the roughness of the pipe wall. It is clear that should be a function of V, D, and The dependence of on the fluid properties and is expected because of the dependence of on these parameters. Although the pressure drop for laminar pipe flow is found to be independent of the roughness of the pipe, it is necessary to include this parameter when considering turbulent flow. As is discussed in Section 8.3.3 and illustrated in Fig. 8.19, for turbulent flow there is a relatively thin viscous sublayer formed in the fluid near the pipe wall. In many instances this layer is very thin; where is the sublayer thickness. If a typical wall roughness element protrudes sufficiently far into 1or even through2 this layer, the structure and properties of the viscous sublayer 1along with and 2 will be different than if the wall were smooth. Thus, for turbulent flow the pressure drop is expected to be a function of the wall roughness. For laminar flow there is no thin viscous layer—viscous effects are important across the entire pipe. Thus, relatively small roughness elements have completely negligible effects on laminar pipe flow. Of course, for pipes with very large wall "roughness" such as that in corrugated pipes, the flowrate may be a function of the "roughness." We will consider only typical constant diameter pipes with relative roughnesses in the range Analysis of flow in corrugated pipes does not fit into the standard constant diameter pipe category, although experimental results for such pipes are available 1Ref. 302. The list of parameters given in Eq. 8.32 is apparently a complete one. That is, experiments have shown that other parameters 1such as surface tension, vapor pressure, etc.2 do not affect the pressure drop for the conditions stated 1steady, incompressible flow; round, horizontal pipe2. Since there are seven variables which can be written in terms of the three reference dimensions MLT Eq. 8.32 can be written in dimensionless form in terms of dimensionless groups. As was discussed in Section 7.9.1, one such representation is This result differs from that used for laminar flow 1see Eq. 8.172 in two ways. First, we have chosen to make the pressure dimensionless by dividing by the dynamic pressure, rather than a characteristic viscous shear stress, This convention was chosen in recognition of the fact that the shear stress for turbulent flow is normally dominated by which is a stronger function

A head loss 1the exit loss2 is also produced when a fluid flows from a pipe into a tank as is shown in Fig. 8.25. In these cases the entire kinetic energy of the exiting fluid 1velocity 2 is dissipated through viscous effects as the stream of fluid mixes with the fluid in the tank and eventually comes to rest The exit loss from points 112 and 122 is therefore equivalent to one velocity head, or Losses also occur because of a change in pipe diameter as is shown in Figs. 8.26 and 8.27. The sharp-edged entrance and exit flows discussed in the previous paragraphs are limiting cases of this type of flow with either respectively. The loss coefficient for a sudden contraction, is a function of the area ratio, as is shown in Fig. 8.26. The value of changes gradually from one extreme of a sharp-edged entrance with to the other extreme of no area change with In many ways, the flow in a sudden expansion is similar to exit flow. As is indicated in Fig. 8.28, the fluid leaves the smaller pipe and initially forms a jet-type structure as it enters the larger pipe. Within a few diameters downstream of the expansion, the jet becomes dispersed across the pipe, and fully developed flow becomes established again. In this process [between sections 122 and 132] a portion of the kinetic energy of the fluid is dissipated as a result of viscous effects. A squareedged exit is the limiting case with A sudden expansion is one of the few components 1perhaps the only one2 for which the loss coefficient can be obtained by means of a simple analysis. To do this we consider the continuity

A typical orifice meter is constructed by inserting between two flanges of a pipe a flat plate with a hole, as shown in Fig. 8.40. The pressure at point 122 within the vena contracta is less than that at point 112. Nonideal effects occur for two reasons. First, the vena contracta area, is less than the area of the hole, by an unknown amount. Thus, where is the contraction coefficient Second, the swirling flow and turbulent motion near the orifice plate introduce a head loss that cannot be calculated theoretically. Thus, an orifice discharge coefficient, is used to take these effects into account. That is, (8.38) where is the area of the hole in the orifice plate. The value of is a function of and the Reynolds number where Typical values of are given in Fig. 8.41. As shown by Eq. 8.38 and the figure in the margin, for a given value of , the flowrate is proportional to the square root of the pressure difference. Note that the value of depends on the specific construction of the orifice meter 1i.e., the placement of the pressure taps, whether the orifice plate edge is square or beveled, etc.2. Very precise conditions governing the construction of standard orifice meters have been established to provide the greatest accuracy possible 1Refs. 23, 242. Another type of pipe flow meter that is based on the same principles used in the orifice meter is the nozzle meter, three variations of which are shown in Fig. 8.42. This device uses a contoured nozzle 1typically placed between flanges of pipe sections2 rather than a simple 1and less expensive2 plate with a hole as in an orifice meter. The resulting flow pattern for the nozzle meter is closer to ideal than the orifice meter flow. There is only a slight vena contracta and the secondary

Although fully developed laminar pipe flow is simple enough to allow the rather straightforward solutions discussed in the previous two sections, it may be worthwhile to consider this flow from a dimensional analysis standpoint. Thus, we assume that the pressure drop in the horizontal pipe, is a function of the average velocity of the fluid in the pipe, V, the length of the pipe, the pipe diameter, D, and the viscosity of the fluid, , as shown by the figure in the margin. We have not included the density or the specific weight of the fluid as parameters because for such flows they are not important parameters. There is neither mass 1density2 times acceleration nor a component of weight 1specific weight times volume2 in the flow direction involved. Thus, There are five variables that can be described in terms of three reference dimensions 1M, L, T2. According to the results of dimensional analysis 1Chapter 72, this flow can be described in terms of dimensionless groups. One such representation is (8.17) where is an unknown function of the length to diameter ratio of the pipe. Although this is as far as dimensional analysis can take us, it seems reasonable to impose a further assumption that the pressure drop is directly proportional to the pipe length. That is, it takes twice the pressure drop to force fluid through a pipe if its length is doubled. The only way that this can be true is if where C is a constant. Thus, Eq. 8.17 becomes which can be rewritten as or (8.18) The basic functional dependence for laminar pipe flow given by Eq. 8.18 is the same as that obtained by the analysis of the two previous sections. The value of C must be determined by theory 1as done in the previous two sections2 or experiment. For a round pipe, For ducts of other cross-sectional shapes, the value of C is different 1see Section 8.4.32. It is usually advantageous to describe a process in terms of dimensionless quantities. To this end we rewrite the pressure drop equation for laminar horizontal pipe flow, Eq. 8.8, as and divide both sides by the dynamic pressure, to obtain the dimensionless form as This is often written as where the dimensionless quantity is termed the friction factor, or sometimes the Darcy friction factor [H. P. G. Darcy (1803-1858)]. 1This parameter should not be confused with the less-used Fanning friction

Another useful pipe flowrate meter is a turbine meter as is shown in Fig. 8.47. A small, freely rotating propeller or turbine within the turbine meter rotates with an angular velocity that is a function of 1nearly proportional to2 the average fluid velocity in the pipe. This angular velocity is picked up magnetically and calibrated to provide a very accurate measure of the flowrate through the meter. 8.6.2 Volume Flow Meters In many instances it is necessary to know the amount 1volume or mass2 of fluid that has passed through a pipe during a given time period, rather than the instantaneous flowrate. For example, we are interested in how many gallons of gasoline are pumped into the tank in our car rather than the rate at which it flows into the tank. There are numerous quantity-measuring devices that provide such information. The nutating disk meter shown in Fig. 8.48 is widely used to measure the net amount of water used in domestic and commercial water systems as well as the amount of gasoline delivered to your gas tank. This meter contains only one essential moving part and is relatively inexpensive and accurate. Its operating principle is very simple, but it may be difficult to understand its operation without actually inspecting the device firsthand. The device consists of a metering chamber with spherical sides and conical top and bottom. A disk passes through a central sphere and divides the chamber into two portions. The disk is constrained to be at an angle not normal to the axis of symmetry of the chamber. A radial plate 1diaphragm2 divides the chamber so that the entering fluid causes the disk to wobble 1nutate2, with fluid flowing alternately above or below the disk. The fluid exits the chamber after the disk has completed one wobble, which corresponds to a specific volume of fluid passing through the chamber. During each wobble of the disk, the pin attached to the tip

Any fluid flowing in a pipe had to enter the pipe at some location. The region of flow near where the fluid enters the pipe is termed the entrance region and is illustrated in Fig. 8.5. It may be the first few feet of a pipe connected to a tank or the initial portion of a long run of a hot air duct coming from a furnace. As is shown in Fig. 8.5, the fluid typically enters the pipe with a nearly uniform velocity profile at section 112. As the fluid moves through the pipe, viscous effects cause it to stick to the pipe wall 1the no-slip boundary condition2. This is true whether the fluid is relatively inviscid air or a very viscous oil. Thus, a boundary layer in which viscous effects are important is produced along the pipe wall such that the initial velocity profile changes with distance along the pipe, x, until the fluid reaches the end of the entrance length, section 122, beyond which the velocity profile does not vary with x. The boundary layer has grown in thickness to completely fill the pipe. Viscous effects are of considerable importance within the boundary layer. For fluid outside the boundary layer [within the inviscid core surrounding the centerline from 112 to 122], viscous effects are negligible. The shape of the velocity profile in the pipe depends on whether the flow is laminar or turbulent, as does the length of the entrance region, As with many other properties of pipe flow, the dimensionless entrance length, correlates quite well with the Reynolds number. Typical entrance lengths are given by (8.1) and (8.2) For very low Reynolds number flows the entrance length can be quite short if whereas for large Reynolds number flows it may take a length equal to many pipe diameters before the end of the entrance region is reached for For many practical engineering problems, so that as shown by the figure in the margin, Calculation of the velocity profile and pressure distribution within the entrance region is quite complex. However, once the fluid reaches the end of the entrance region, section 122 of Fig. 8.5, the flow is simpler to describe because the velocity is a function of only the distance from the pipe centerline, r, and independent of x. This is true until the character of the pipe changes in some way, such as a change in diameter, or the fluid flows through a bend, valve, or some other component at section 132. The flow between 122 and 132 is termed fully developed flow. Beyond the interruption of the fully developed flow [at section 142], the flow gradually begins its

As discussed in the previous section, the head loss in long, straight sections of pipe, the major losses, can be calculated by use of the friction factor obtained from either the Moody chart or the Colebrook equation. Most pipe systems, however, consist of considerably more than straight pipes. These additional components 1valves, bends, tees, and the like2 add to the overall head loss of the system. Such losses are generally termed minor losses, with the corresponding head loss denoted In this section we indicate how to determine the various minor losses that commonly occur in pipe systems. The head loss associated with flow through a valve is a common minor loss. The purpose of a valve is to provide a means to regulate the flowrate. This is accomplished by changing the geometry of the system 1i.e., closing or opening the valve alters the flow pattern through the valve2, which in turn alters the losses associated with the flow through the valve. The flow resistance or head loss through the valve may be a significant portion of the resistance in the system. In fact, with the valve closed, the resistance to the flow is infinite—the fluid cannot flow. Such minor losses may be very important indeed. With the valve wide open the extra resistance due to the presence of the valve may or may not be negligible. The flow pattern through a typical component such as a valve is shown in Fig. 8.21. It is not difficult to realize that a theoretical analysis to predict the details of such flows to obtain the head loss for these components is not, as yet, possible. Thus, the head loss information for essentially all components is given in dimensionless form and based on experimental data. The most common method used to determine these head losses or pressure drops is to specify the loss coefficient, which is defined as

As is indicated in Eq. 8.8, the average velocity is one-half of the maximum velocity. In general, for velocity profiles of other shapes 1such as for turbulent pipe flow2, the average velocity is not merely the average of the maximum and minimum 102 velocities as it is for the laminar parabolic profile. The two velocity profiles indicated in Fig. 8.9 provide the same flowrate—one is the fictitious ideal profile; the other is the actual laminar flow profile. The above results confirm the following properties of laminar pipe flow. For a horizontal pipe the flowrate is 1a2 directly proportional to the pressure drop, 1b2 inversely proportional to the viscosity, 1c2 inversely proportional to the pipe length, and 1d2 proportional to the pipe diameter to the fourth power. With all other parameters fixed, an increase in diameter by a factor of 2 will increase the flowrate by a factor of 24 16—the flowrate is very strongly dependent on pipe size. This dependence is shown by the figure in the margin. Likewise, a small error in pipe diameter can cause a relatively large error in flowrate. For example, a 2% error in diameter gives an 8% error in flowrate or so that This flow, the properties of which were first established experimentally by two independent workers, G. Hagen 11797-18842 in 1839 and J. Poiseuille 11799-18692 in 1840, is termed Hagen-Poiseuille flow. Equation 8.9 is commonly referred to as Poiseuille's law. Recall that all of these results are restricted to laminar flow 1those with Reynolds numbers less than approximately 21002 in a horizontal pipe. The adjustment necessary to account for nonhorizontal pipes, as shown in Fig. 8.10, can be easily included by replacing the pressure drop, by the combined effect of pressure and gravity, , where is the angle between the pipe and the horizontal. 1Note that if the flow is uphill, while if the flow is downhill.2 This can be seen from the force balance in the x direction 1along the pipe axis2 on the cylinder of fluid shown in Fig. 8.10b. The method is exactly analogous to that used to obtain the Bernoulli equation 1Eq. 3.62 when the streamline is not horizontal. The net force in the x direction is a combination of the pressure force in that direction, and the component of weight in that direction, The result is a slightly modified form of Eq. 8.3 given by (8.10) Thus, all of the results for the horizontal pipe are valid provided the pressure gradient is adjusted for the elevation term, that is, is replaced by so that (8.11) and (8.12) It is seen that the driving force for pipe flow can be either a pressure drop in the flow direction, or the component of weight in the flow direction, If the flow is downhill, gravity helps the flow 1a smaller pressure drop is required; 2. If the flow is uphill, gravity works against the flow 1a larger pressure drop is required; sin u 7 02. Note that g/ sin u g¢z 1where

As is indicated in the previous section, the flow in long, straight, constant diameter sections of a pipe becomes fully developed. That is, the velocity profile is the same at any cross section of the pipe. Although this is true whether the flow is laminar or turbulent, the details of the velocity profile 1and other flow properties2 are quite different for these two types of flow. As will be seen in the remainder of this chapter, knowledge of the velocity profile can lead directly to other useful information such as pressure drop, head loss, flowrate, and the like. Thus, we begin by developing the equation for the velocity profile in fully developed laminar flow. If the flow is not fully developed, a theoretical analysis becomes much more complex and is outside the scope of this text. If the flow is turbulent, a rigorous theoretical analysis is as yet not possible. Although most flows are turbulent rather than laminar, and many pipes are not long enough to allow the attainment of fully developed flow, a theoretical treatment and full understanding of fully developed laminar flow is of considerable importance. First, it represents one of the few theoretical viscous analyses that can be carried out "exactly" 1within the framework of quite general assumptions2 without using other ad hoc assumptions or approximations. An understanding of the method of analysis and the results obtained provides a foundation from which to carry out more complicated analyses. Second, there are many practical situations involving the use of fully developed laminar pipe flow. There are numerous ways to derive important results pertaining to fully developed laminar flow. Three alternatives include: 112 from applied directly to a fluid element, 122 from the Navier -Stokes equations of motion, and 132 from dimensional analysis methods.

Before we apply the various governing equations to pipe flow examples, we will discuss some of the basic concepts of pipe flow. With these ground rules established we can then proceed to formulate and solve various important flow problems. Although not all conduits used to transport fluid from one location to another are round in cross section, most of the common ones are. These include typical water pipes, hydraulic hoses, and other conduits that are designed to withstand a considerable pressure difference across their walls without undue distortion of their shape. Typical conduits of noncircular cross section include heating and air conditioning ducts that are often of rectangular cross section. Normally the pressure difference between the inside and outside of these ducts is relatively small. Most of the basic principles involved are independent of the cross-sectional shape, although the details of the flow may be dependent on it. Unless otherwise specified, we will assume that the conduit is round, although we will show how to account for other shapes.

Chaos theory is a relatively new branch of mathematical physics that may provide insight into the complex nature of turbulence. This method combines mathematics and numerical 1computer2 techniques to provide a new way to analyze certain problems. Chaos theory, which is quite complex and is currently under development, involves the behavior of nonlinear dynamical systems and their response to initial and boundary conditions. The flow of a viscous fluid, which is governed by the nonlinear Navier- Stokes equations 1Eq. 6.1272, may be such a system. To solve the Navier-Stokes equations for the velocity and pressure fields in a viscous flow, one must specify the particular flow geometry being considered 1the boundary conditions2 and the condition of the flow at some particular time 1the initial conditions2. If, as some researchers predict, the Navier-Stokes equations allow chaotic behavior, then the state of the flow at times after the initial time may be very, very sensitive to the initial conditions. A slight variation to the initial flow conditions may cause the flow at later times to be quite different than it would have been with the original, only slightly different initial conditions. When carried to the extreme, the flow may be "chaotic," "random," or perhaps 1in current terminology2, "turbulent." The occurrence of such behavior would depend on the value of the Reynolds number. For example, it may be found that for sufficiently small Reynolds numbers the flow is not chaotic 1i.e., it is laminar2, while for large Reynolds numbers it is chaotic with turbulent characteristics. Thus, with the advancement of chaos theory it may be found that the numerous ad hoc turbulence ideas mentioned in previous sections 1i.e., eddy viscosity, mixing length, law of the wall, etc.2 may not be needed. It may be that chaos theory can provide the turbulence properties and structure directly from the governing equations. As of now we must wait until this exciting topic is developed further. The interested reader is encouraged to consult Ref. 4 for a general introduction to chaos or Ref. 33 for additional material.

For all flows involved in this chapter, we assume that the pipe is completely filled with the fluid being transported as is shown in Fig. 8.2a. Thus, we will not consider a concrete pipe through which rainwater flows without completely filling the pipe, as is shown in Fig. 8.2b. Such flows, called open-channel flow, are treated in Chapter 10. The difference between open-channel flow and the pipe flow of this chapter is in the fundamental mechanism that drives the flow. For open-channel flow, gravity alone is the driving force—the water flows down a hill. For pipe flow, gravity may be important 1the pipe need not be horizontal2, but the main driving force is likely to be a pressure gradient along the pipe. If the pipe is not full, it is not possible to maintain this pressure difference

For turbulent flow the predominant component of velocity is also along the pipe, but it is unsteady 1random2 and accompanied by random components normal to the pipe axis, Such motion in a typical flow occurs too fast for our eyes to follow. Slow motion pictures of the flow can more clearly reveal the irregular, random, turbulent nature of the flow. As was discussed in Chapter 7, we should not label dimensional quantities as being "large" or "small," such as "small enough flowrates" in the preceding paragraphs. Rather, the appropriate dimensionless quantity should be identified and the "small" or "large" character attached to it. A quantity is "large" or "small" only relative to a reference quantity. The ratio of those quantities results in a dimensionless quantity. For pipe flow the most important dimensionless parameter is the Reynolds number, Re—the ratio of the inertia to viscous effects in the flow. Hence, in the previous paragraph the term flowrate should be replaced by Reynolds number, where V is the average velocity in the pipe. That is, the flow in a pipe is laminar, transitional, or turbulent provided the Reynolds number is "small enough," "intermediate," or "large enough." It is not only the fluid velocity that determines the character of the flow—its density, viscosity, and the pipe size are of equal importance. These parameters combine to produce the Reynolds number. The distinction between laminar and turbulent pipe flow and its dependence on an appropriate dimensionless quantity was first pointed out by Osborne Reynolds in 1883. The Reynolds number ranges for which laminar, transitional, or turbulent pipe flows are obtained cannot be precisely given. The actual transition from laminar to turbulent flow may take place at various Reynolds numbers, depending on how much the flow is disturbed by vibrations of the pipe, roughness of the entrance region, and the like. For general engineering purposes 1i.e., without undue precautions to eliminate such disturbances2, the following values are appropriate: The flow in a round pipe is laminar if the Reynolds number is less than approximately 2100. The flow in a round pipe is turbulent if the Reynolds number is greater than approximately 4000. For Reynolds numbers between these two limits, the flow may switch between laminar and turbulent conditions in an apparently random fashion 1transitional flow2

In many pipe systems there is more than one pipe involved. The complex system of tubes in our lungs 1beginning as shown by the figure in the margin, with the relatively large-diameter trachea and ending in tens of thousands of minute bronchioles after numerous branchings2 and the maze of pipes in a city's water distribution system are typical of such systems. The governing mechanisms for the flow in multiple pipe systems are the same as for the single pipe systems discussed in this chapter. However, because of the numerous unknowns involved, additional complexities may arise in solving for the flow in multiple pipe systems. Some of these complexities are discussed in this section.

In the previous chapters we have considered a variety of topics concerning the motion of fluids. The basic governing principles concerning mass, momentum, and energy were developed and applied, in conjunction with rather severe assumptions, to numerous flow situations. In this chapter we will apply the basic principles to a specific, important topic—the incompressible flow of viscous fluids in pipes and ducts. The transport of a fluid 1liquid or gas2 in a closed conduit 1commonly called a pipe if it is of round cross section or a duct if it is not round2 is extremely important in our daily operations. A brief consideration of the world around us will indicate that there is a wide variety of applications of pipe flow. Such applications range from the large, man-made Alaskan pipeline that carries crude oil almost 800 miles across Alaska, to the more complex 1and certainly not less useful2 natural systems of "pipes" that carry blood throughout our body and air into and out of our lungs. Other examples

In the previous section various properties of fully developed laminar pipe flow were discussed. Since turbulent pipe flow is actually more likely to occur than laminar flow in practical situations, it is necessary to obtain similar information for turbulent pipe flow. However, turbulent flow is a very complex process. Numerous persons have devoted considerable effort in attempting to understand the variety of baffling aspects of turbulence. Although a considerable amount of knowledge about the topic has been developed, the field of turbulent flow still remains the least understood area of fluid mechanics. In this book we can provide only some of the very basic ideas concerning turbulence. The interested reader should consult some of the many books available for further reading 1Refs. 1-32. 8.3.1 Transition from Laminar to Turbulent Flow Flows are classified as laminar or turbulent. For any flow geometry, there is one 1or more2 dimensionless parameter such that with this parameter value below a particular value the flow is laminar, whereas with the parameter value larger than a certain value the flow is turbulent. The important parameters involved 1i.e., Reynolds number, Mach number2 and their critical values depend on the specific flow situation involved. For example, flow in a pipe and flow along a flat plate 1boundary layer flow, as is discussed in Section 9.2.42 can be laminar or turbulent, depending on the value of the Reynolds number involved. As a general rule for pipe flow, the value of the Reynolds number must be less than approximately 2100 for laminar flow and greater than approximately 4000 for turbulent flow. For flow along a flat plate the transition between laminar and turbulent flow occurs at a Reynolds number of approximately 500,000 1see Section 9.2.42, where the length term in the Reynolds number is the distance measured from the leading edge of the plate. Consider a long section of pipe that is initially filled with a fluid at rest. As the valve is opened to start the flow, the flow velocity and, hence, the Reynolds number increase from zero 1no flow2 to their maximum steady-state flow values, as is shown in Fig. 8.11. Assume this transient process is slow enough so that unsteady effects are negligible 1quasi-steady flow2. For an initial time period the Reynolds number is small enough for laminar flow to occur. At some time the Reynolds number reaches 2100, and the flow begins its transition to turbulent conditions. Intermittent spots or bursts of turbulence appear. As the Reynolds number is increased, the entire flow field becomes turbulent. The flow remains turbulent as long as the Reynolds number exceeds approximately 4000. A typical trace of the axial component of velocity measured at a given location in the flow, is shown in Fig. 8.12. Its irregular, random nature is the distinguishing feature of turbulent flow. The character of many of the important properties of the flow 1pressure drop, heat transfer, etc.2 depends strongly on the existence and nature of the turbulent fluctuations or randomness

In the previous section we obtained results for fully developed laminar pipe flow by applying Newton's second law and the assumption of a Newtonian fluid to a specific portion of the fluid— a cylinder of fluid centered on the axis of a long, round pipe. When this governing law and assumptions are applied to a general fluid flow 1not restricted to pipe flow2, the result is the Navier -Stokes equations as discussed in Chapter 6. In Section 6.9.3 these equations were solved for the specific geometry of fully developed laminar flow in a round pipe. The results are the same as those given in Eq. 8.7.

In the previous sections of this chapter, we discussed concepts concerning flow in pipes and ducts. The purpose of this section is to apply these ideas to the solutions of various practical problems. The application of the pertinent equations is straightforward, with rather simple calculations that give answers to problems of engineering importance. The main idea involved is to apply the energy equation between appropriate locations within the flow system, with the head loss written in terms of the friction factor and the minor loss coefficients. We will consider two classes of pipe systems: those containing a single pipe 1whose length may be interrupted by various components2, and those containing multiple pipes in parallel, series, or network configurations.

It is often necessary to determine experimentally the flowrate in a pipe. In Chapter 3 we introduced various types of flow-measuring devices 1Venturi meter, nozzle meter, orifice meter, etc.2 and discussed their operation under the assumption that viscous effects were not important. In this section we will indicate how to account for the ever-present viscous effects in these flow meters. We will also indicate other types of commonly used flow meters. 8.6.1 Pipe Flowrate Meters Three of the most common devices used to measure the instantaneous flowrate in pipes are the orifice meter, the nozzle meter, and the Venturi meter. As was discussed in Section 3.6.3, each of these meters operates on the principle that a decrease in flow area in a pipe causes an increase in velocity that is accompanied by a decrease in pressure. Correlation of the pressure difference with the velocity provides a means of measuring the flowrate. In the absence of viscous effects and under the assumption of a horizontal pipe, application of the Bernoulli equation 1Eq. 3.72 between points 112 and 122 shown in Fig. 8.39 gave (8.37) where Based on the results of the previous sections of this chapter, we anticipate that there is a head loss between 112 and 122 so that the governing equations become and The ideal situation has and results in Eq. 8.37. The difficulty in including the head loss is that there is no accurate expression for it. The net result is that empirical coefficients are used in the flowrate equations to account for the complex real-world effects brought on by the nonzero viscosity. The coefficients are discussed in this section.

Many of the conduits that are used for conveying fluids are not circular in cross section. Although the details of the flows in such conduits depend on the exact cross-sectional shape, many round pipe results can be carried over, with slight modification, to flow in conduits of other shapes. Theoretical results can be obtained for fully developed laminar flow in noncircular ducts, although the detailed mathematics often becomes rather cumbersome. For an arbitrary 8.4 Dimensional Analysis of Pipe Flow 425 wind tunnel diffuser is interrupted by the four turning corners and the fan, it may not be possible to obtain a smaller value of for this situation. Thus, The loss coefficients for the conical nozzle between section 142 and 152 and the flow-straightening screens are assumed to be and 1Ref. 132, respectively. We neglect the head loss in the relatively short test section. Thus, the total head loss is or or Hence, from Eq. 1 we obtain the pressure rise across the fan as (Ans) From Eq. 2 we obtain the power added to the fluid as or p (Ans) a 34,300 ft # lbs 550 1ft # lbs2hp 62.3 hp 34,300 ft # lbs pa 10.0765 lbft3 214.0 ft2 21200 fts21560 ft2 42.8 lbft2 0.298 psi p1 p9 ghL19 10.0765 lbft3 21560 ft2 hL19 560 ft 0.212002 2 4.0122.92 2 4 ft2 s 2 32132.2 fts 2 2 4 30.2180.02 44.42 28.62 22.92 2 0.612002 2 0.6V2 6 0.2V2 5 4.0V2 4 42g hL19 30.21V2 7 V2 8 V2 2 V2 32 hLdif hLnoz hLscr hL19 hLcorner7 hLcorner8 hLcorner2 hLcorner3 KL KLscr 4.0 noz 0.2 hLdif KLdif V2 6 2g 0.6 V2 6 2g KLdif COMMENTS By repeating the calculations with various test section velocities, , the results shown in Fig. E8.6c are obtained. Since the head loss varies as and the power varies as head loss times , it follows that the power varies as the cube of the velocity. Thus, doubling the wind tunnel speed requires an eightfold increase in power. With a closed-return wind tunnel of this type, all of the power required to maintain the flow is dissipated through viscous effects, with the energy remaining within the closed tunnel. If heat transfer across the tunnel walls is negligible, the air temperature within the tunnel will increase in time. For steadystate operations of such tunnels, it is often necessary to provide some means of cooling to maintain the temperature at acceptable levels. It should be noted that the actual size of the motor that powers the fan must be greater than the calculated 62.3 hp because the fan is not 100% efficient. The power calculated above is that needed by the fluid to overcome losses in the tunnel, excluding those in the fan. If the fan were 60% efficient, it would require a shaft power of to run the fan. Determination of fan 1or pump2 efficiencies can be a complex problem that depends on the specific geometry of the fan. Introductory material about fan performance is presented in Chapter 12; additional material can be found in various references 1Refs. 14, 15, 16, for example2. It should also be noted that the above results are only approximate. Clever, careful design of the various components 1corners, diffuser, etc.2 may lead to improved 1i.e., lower2 values of the various loss coefficients, and hence lower power requirements. Since is proportional to the components with the larger V tend to have the larger head loss. Thus, even though for each of the four corners, the head loss for corner 172 is times greater than it is for corner 132. 1V7V32 2 18022.92 2 12.2 KL 0.2 V2 hL , p 62.3 hp10.602 104 hp V5 V5 2 V5 TABLE E8.6 Location Area ( ) Velocity ( ) 1 22.0 36.4 2 28.0 28.6 3 35.0 22.9 4 35.0 22.9 5 4.0 200.0 6 4.0 200.0 7 10.0 80.0 8 18.0 44.4 9 22.0 36.4 ft fts 2 (200 ft/s, 62.3 hp) 250 200 150 100 50 0 0 50 100 150 V5, ft/s a, hp 200 250 300 F I G U R E E8.6c JWCL068_ch08_383-460.qxd 9/23/08 10:54 AM Page 425 cross section, as is shown in Fig. 8.34, the velocity profile is a function of both y and z This means that the governing equation from which the velocity profile is obtained 1either the Navier-Stokes equations of motion or a force balance equation similar to that used for circular pipes, Eq. 8.62 is a partial differential equation rather than an ordinary differential equation. Although the equation is linear 1for fully developed flow the convective acceleration is zero2, its solution is not as straightforward as for round pipes. Typically the velocity profile is given in terms of an infinite series representation 1Ref. 172. Practical, easy-to-use results can be obtained as follows. Regardless of the cross-sectional shape, there are no inertia effects in fully developed laminar pipe flow. Thus, the friction factor can be written as where the constant C depends on the particular shape of the duct, and is the Reynolds number, based on the hydraulic diameter. The hydraulic diameter defined as is four times the ratio of the cross-sectional flow area divided by the wetted perimeter, P, of the pipe as is illustrated in Fig. 8.34. It represents a characteristic length that defines the size of a cross section of a specified shape. The factor of 4 is included in the definition of so that for round pipes the diameter and hydraulic diameter are equal The hydraulic diameter is also used in the definition of the friction factor, and the relative roughness, The values of for laminar flow have been obtained from theory and or experiment for various shapes. Typical values are given in Table 8.3 along with the hydraulic diameter. Note

Newtonian fluid, the shear stress is simply proportional to the velocity gradient, 1see Section 1.62. In the notation associated with our pipe flow, this becomes (8.6) The negative sign is included to give with 1the velocity decreases from the pipe centerline to the pipe wall2. Equations 8.3 and 8.6 represent the two governing laws for fully developed laminar flow of a Newtonian fluid within a horizontal pipe. The one is Newton's second law of motion and the other is the definition of a Newtonian fluid. By combining these two equations we obtain which can be integrated to give the velocity profile as follows: or where is a constant. Because the fluid is viscous it sticks to the pipe wall so that at Thus, Hence, the velocity profile can be written as (8.7) where is the centerline velocity. An alternative expression can be written by using the relationship between the wall shear stress and the pressure gradient 1Eqs. 8.5 and 8.72 to give where is the pipe radius. This velocity profile, plotted in Fig. 8.9, is parabolic in the radial coordinate, r, has a maximum velocity, at the pipe centerline, and a minimum velocity 1zero2 at the pipe wall. The volume flowrate through the pipe can be obtained by integrating the velocity profile across the pipe. Since the flow is axisymmetric about the centerline, the velocity is constant on small area elements consisting of rings of radius r and thickness dr as shown in the figure in the margin. Thus, or By definition, the average velocity is the flowrate divided by the cross-sectional area, so that for this flow

Numerous other devices are used to measure the flowrate in pipes. Many of these devices use principles other than the high-speed/low-pressure concept of the orifice, nozzle, and Venturi meters. A quite common, accurate, and relatively inexpensive flow meter is the rotameter, or variable area meter as is shown in Fig. 8.46. In this device a float is contained within a tapered, transparent metering tube that is attached vertically to the pipeline. As fluid flows through the meter 1entering at the bottom2, the float will rise within the tapered tube and reach an equilibrium height that is a function of the flowrate. This height corresponds to an equilibrium condition for which the net force on the float 1buoyancy, float weight, fluid drag2 is zero. A calibration scale in the tube provides the relationship between the float position and the flowrate.

Pipe flow problems in which it is desired to determine the flowrate for a given set of conditions 1Type II problems2 often require trial-and-error or numerical root-finding techniques. This is because it is necessary to know the value of the friction factor to carry out the calculations, but the friction factor is a function of the unknown velocity 1flowrate2 in terms of the Reynolds number. The solution procedure is indicated in Example 8.10. In pipe flow problems for which the diameter is the unknown 1Type III2, an iterative or numerical root-finding technique is required. This is, again, because the friction factor is a function of the diameter—through both the Reynolds number and the relative roughness. Thus, neither 4rQpmD nor eD are known unless D is known. Examples 8.12 and 8.13 illustrate this.

The flow of a fluid in a pipe may be laminar flow or it may be turbulent flow. Osborne Reynolds 11842-19122, a British scientist and mathematician, was the first to distinguish the difference between these two classifications of flow by using a simple apparatus as shown by the figure in the margin, which is a sketch of Reynolds' dye experiment. Reynolds injected dye into a pipe in which water flowed due to gravity. The entrance region of the pipe is depicted in Fig. 8.3a. If water runs through a pipe of diameter D with an average velocity V, the following characteristics are observed by injecting neutrally buoyant dye as shown. For "small enough flowrates" the dye streak 1a streakline2 will remain as a well-defined line as it flows along, with only slight blurring due to molecular diffusion of the dye into the surrounding water. For a somewhat larger "intermediate flowrate" the dye streak fluctuates in time and space, and intermittent bursts of irregular behavior appear along the streak. On the other hand, for "large enough flowrates" the dye streak almost immediately becomes blurred and spreads across the entire pipe in a random fashion. These three characteristics, denoted as laminar, transitional, and turbulent flow, respectively, are illustrated in Fig. 8.3b. The curves shown in Fig. 8.4 represent the x component of the velocity as a function of time at a point A in the flow. The random fluctuations of the turbulent flow 1with the associated particle mixing2 are what disperse the dye throughout the pipe and cause the blurred appearance illustrated in Fig. 8.3b. For laminar flow in a pipe there is only one component of velocity,

The fluctuations are equally distributed on either side of the average. It is also clear, as is indicated in Fig. 8.13, that since the square of a fluctuation quantity cannot be negative its average value is positive. Thus, On the other hand, it may be that the average of products of the fluctuations, such as are zero or nonzero 1either positive or negative2. The structure and characteristics of turbulence may vary from one flow situation to another. For example, the turbulence intensity 1or the level of the turbulence2 may be larger in a very gusty wind than it is in a relatively steady 1although turbulent2 wind. The turbulence intensity, is often defined as the square root of the mean square of the fluctuating velocity divided by the timeaveraged velocity, or The larger the turbulence intensity, the larger the fluctuations of the velocity 1and other flow parameters2. Well-designed wind tunnels have typical values of although with extreme care, values as low as have been obtained. On the other hand, values of are found for the flow in the atmosphere and rivers. A typical atmospheric wind speed graph is shown in the figure in the margin. Another turbulence parameter that is different from one flow situation to another is the period of the fluctuations—the time scale of the fluctuations shown in Fig. 8.12. In many flows, such as the flow of water from a faucet, typical frequencies are on the order of 10, 100, or 1000 cycles per second 1cps2. For other flows, such as the Gulf Stream current in the Atlantic Ocean or flow of the atmosphere of Jupiter, characteristic random oscillations may have a period on the order of hours, days, or more. It is tempting to extend the concept of viscous shear stress for laminar flow to that of turbulent flow by replacing u, the instantaneous velocity, by the time-averaged velocity. However, numerous experimental and theoretical studies have shown that such an approach leads to completely incorrect results. That is, A physical explanation for this behavior can be found in the concept of what produces a shear stress. Laminar flow is modeled as fluid particles that flow smoothly along in layers, gliding past the slightly slower or faster ones on either side. As is discussed in Chapter 1, the fluid actually consists of numerous molecules darting about in an almost random fashion as is indicated in Fig. 8.14a. The motion is not entirely random—a slight bias in one direction produces the flowrate we associate

The following equation from Colebrook is valid for the entire nonlaminar range of the Moody chart (8.35a) In fact, the Moody chart is a graphical representation of this equation, which is an empirical fit of the pipe flow pressure drop data. Equation 8.35 is called the Colebrook formula. A difficulty with its use is that it is implicit in the dependence of f. That is, for given conditions it is not possible to solve for f without some sort of iterative scheme. With the use of modern computers and calculators, such calculations are not difficult. A word of caution is in order concerning the use of the Moody chart or the equivalent Colebrook formula. Because of various inherent inaccuracies involved 1uncertainty in the relative roughness, uncertainty in the experimental data used to produce the Moody chart, etc.2, the use of several place accuracy in pipe flow problems is usually not justified. As a rule of thumb, a 10% accuracy is the best expected. It is possible to obtain an equation that adequately approximates the ColebrookMoody chart relationship but does not require an iterative scheme. For example, an alternate form (Ref. 34), which is easier to use, is given by (8.35b) where one can solve for f explicitly

The nature of the solution process for pipe flow problems can depend strongly on which of the various parameters are independent parameters 1the "given"2 and which is the dependent parameter 1the "determine"2. The three most common types of problems are shown in Table 8.4 in terms of the parameters involved. We assume the pipe system is defined in terms of the length of pipe sections used and the number of elbows, bends, and valves needed to convey the fluid between the desired locations. In all instances we assume the fluid properties are given. In a Type I problem we specify the desired flowrate or average velocity and determine the necessary pressure difference or head loss. For example, if a flowrate of 2.0 galmin is required for a dishwasher that is connected to the water heater by a given pipe system as shown by the figure in the margin, what pressure is needed in the water heater? In a Type II problem we specify the applied driving pressure 1or, alternatively, the head loss2 and determine the flowrate. For example, how many galmin of hot water are supplied to the dishwasher if the pressure within the water heater is 60 psi and the pipe system details 1length, diameter, roughness of the pipe; number of elbows; etc.2 are specified

The pressure drop across a component that has a loss coefficient of is equal to the dynamic pressure, As shown by Eq. 8.36 and the figure in the margin, for a given value of KL the head loss is proportional to the square of the velocity. The actual value of is strongly dependent on the geometry of the component considered. It may also be dependent on the fluid properties. That is, where is the pipe Reynolds number. For many practical applications the Reynolds number is large enough so that the flow through the component is dominated by inertia effects, with viscous effects being of secondary importance. This is true because of the relatively large accelerations and decelerations experienced by the fluid as it flows along a rather curved, variable area 1perhaps even torturous2 path through the component 1see Fig. 8.212. In a flow that is dominated by inertia effects rather than viscous effects, it is usually found that pressure drops and head losses correlate directly with the dynamic pressure. This is the reason why the friction factor for very large Reynolds number, fully developed pipe flow is independent of the Reynolds number. The same condition is found to be true for flow through pipe components. Thus, in most cases of practical interest the loss coefficients for components are a function of geometry only, Minor losses are sometimes given in terms of an equivalent length, In this terminology, the head loss through a component is given in terms of the equivalent length of pipe that would produce the same head loss as the component. That is,

The random velocity components that account for this momentum transfer 1hence, the shear force2 are 1for the x component of velocity2 and 1for the rate of mass transfer crossing the plane2. A more detailed consideration of the processes involved will show that the apparent shear stress on plane A- A is given by the following 1Ref. 22: (8.26) Note that if the flow is laminar, so that and Eq. 8.26 reduces to the customary random molecule-motion-induced laminar shear stress, For turbulent flow it is found that the turbulent shear stress, is positive. Hence, the shear stress is greater in turbulent flow than in laminar flow. Note the units on are or as expected. Terms of the form 1or etc.2 are called Reynolds stresses in honor of Osborne Reynolds who first discussed them in 1895. It is seen from Eq. 8.26 that the shear stress in turbulent flow is not merely proportional to the gradient of the time-averaged velocity, It also contains a contribution due to the random fluctuations of the x and y components of velocity. The density is involved because of the momentum transfer of the fluid within the random eddies. Although the relative magnitude of compared to is a complex function dependent on the specific flow involved, typical measurements indicate the structure shown in Fig. 8.15a. 1Recall from Eq. 8.4 that the shear stress is proportional to the distance from the centerline of the pipe.2 In a very narrow region near the wall 1the viscous sublayer2, the laminar shear stress is dominant. Away from the wall 1in the outer layer2 the turbulent portion of the shear stress is dominant. The transition between these two regions occurs in the overlap layer. The corresponding typical velocity profile is shown in Fig. 8.15b. The scale of the sketches shown in Fig. 8.15 is not necessarily correct. Typically the value of is 100 to 1000 times greater than in the outer region, while the converse is true in the viscous sublayer. A correct modeling of turbulent flow is strongly dependent on an accurate knowledge of This, in turn, requires an accurate knowledge of the fluctuations and or As yet it is not possible to solve the governing equations 1the Navier-Stokes equations2 for these details of the flow, although numerical techniques 1see Appendix A2 using the largest and fastest computers available have produced important information about some of the characteristics of turbulence. Considerable effort has gone into the study of turbulence. Much remains to be learned. Perhaps studies in the new areas of chaos and fractal geometry will provide the tools for a better understanding of turbulence 1see Section 8.3.52. The vertical scale of Fig. 8.15 is also distorted. The viscous sublayer is usually a very thin layer adjacent to the wall. For example, for water flow in a 3-in.-diameter pipe with an average velocity of the viscous sublayer is approximately 0.002 in. thick. Since the fluid motion within this thin layer is critical in terms of the overall flow 1the no-slip condition and the wall shear stress occur in this layer2, it is not surprising to find that turbulent pipe flow properties can be quite

The simplest multiple pipe systems can be classified into series or parallel flows, as are shown in Fig. 8.35. The nomenclature is similar to that used in electrical circuits. Indeed, an analogy between fluid and electrical circuits is often made as follows. In a simple electrical circuit, there is a balance between the voltage 1e2, current 1i2, and resistance 1R2 as given by Ohm's law: In a fluid circuit there is a balance between the pressure drop the flowrate or velocity 1Q or V2, and the flow resistance as given in terms of the friction factor and minor loss coefficients . For a simple flow it follows that where a measure of the resistance to the flow, is proportional to f. The main differences between the solution methods used to solve electrical circuit problems and those for fluid circuit problems lie in the fact that Ohm's law is a linear equation 1doubling the voltage doubles the current2, while the fluid equations are generally nonlinear 1doubling the pressure drop does not double the flowrate unless the flow is laminar2. Thus, although some of the

The turbulent flow characteristics discussed in this section are not unique to turbulent flow in round pipes. Many of the characteristics introduced 1i.e., the Reynolds stress, the viscous sublayer, the overlap layer, the outer layer, the general characteristics of the velocity profile, etc.2 are found in other turbulent flows. In particular, turbulent pipe flow and turbulent flow past a solid wall 1boundary layer flow2 share many of these common traits. Such ideas are discussed more fully in Chapter 9. JWCL068_ch08_383-460.qxd 9/23/08 10:52 AM Page 408 8.3.4 Turbulence Modeling Although it is not yet possible to theoretically predict the random, irregular details of turbulent flows, it would be useful to be able to predict the time-averaged flow fields 1pressure, velocity, etc.2 directly from the basic governing equations. To this end one can time average the governing Navier- Stokes equations 1Eqs. 6.31 and 6.1272 to obtain equations for the average velocity and pressure. However, because the Navier-Stokes equations are nonlinear, the resulting time-averaged differential equations contain not only the desired average pressure and velocity as variables, but also averages of products of the fluctuations—terms of the type that one tried to eliminate by averaging the equations! For example, the Reynolds stress 1see Eq. 8.262 occurs in the timeaveraged momentum equation. Thus, it is not possible to merely average the basic differential equations and obtain governing equations involving only the desired averaged quantities. This is the reason for the variety of ad hoc assumptions that have been proposed to provide "closure" to the equations governing the average flow. That is, the set of governing equations must be a complete or closed set of equations—the same number of equation as unknowns. Various attempts have been made to solve this closure problem 1Refs. 1, 322. Such schemes involving the introduction of an eddy viscosity or the mixing length 1as introduced in Section 8.3.22 are termed algebraic or zero-equation models. Other methods, which are beyond the scope of this book, include the one-equation model and the two-equation model. These turbulence models are based on the equation for the turbulence kinetic energy and require significant computer usage. Turbulence modeling is an important and extremely difficult topic. Although considerable progress has been made, much remains to be done in this area.

The ultimate in multiple pipe systems is a network of pipes such as that shown in Fig. 8.38. Networks like these often occur in city water distribution systems and other systems that may have multiple "inlets" and "outlets." The direction of flow in the various pipes is by no means obvious—in fact, it may vary in time, depending on how the system is used from time to time. The solution for pipe network problems is often carried out by use of node and loop equations similar in many ways to that done in electrical circuits. For example, the continuity equation requires that for each node 1the junction of two or more pipes2 the net flowrate is zero. What flows into a node must flow out at the same rate. In addition, the net pressure difference completely around a loop 1starting at one location in a pipe and returning to that location2 must be zero. By combining these ideas with the usual head loss and pipe flow equations, the flow throughout the entire network can be obtained. Of course, trial-and-error solutions are usually required because the direction of flow and the friction factors may not be known. Such a solution procedure using matrix techniques is ideally suited for computer use 1Refs. 21, 222.

Thus, the flowrate through a Venturi meter is given by (8.40) where is the throat area. The range of values of , the Venturi discharge coefficient, is given in Fig. 8.45. The throat-to-pipe diameter ratio the Reynolds number, and the shape of the converging and diverging sections of the meter are among the parameters that affect the value of Again, the precise values of and depend on the specific geometry of the devices used. Considerable information concerning the design, use, and installation of standard flow meters can be found in various books 1Refs. 23, 24, 25, 26, 312.

Turbulence is also of importance in the mixing of fluids. Smoke from a stack would continue for miles as a ribbon of pollutant without rapid dispersion within the surrounding air if the flow were laminar rather than turbulent. Under certain atmospheric conditions this is observed to occur. Although there is mixing on a molecular scale 1laminar flow2, it is several orders of magnitude slower and less effective than the mixing on a macroscopic scale 1turbulent flow2. It is considerably easier to mix cream into a cup of coffee 1turbulent flow2 than to thoroughly mix two colors of a viscous paint 1laminar flow2. In other situations laminar 1rather than turbulent2 flow is desirable. The pressure drop in pipes 1hence, the power requirements for pumping2 can be considerably lower if the flow is laminar rather than turbulent. Fortunately, the blood flow through a person's arteries is normally laminar, except in the largest arteries with high blood flowrates. The aerodynamic drag on an airplane wing can be considerably smaller with laminar flow past it than with turbulent flow. 8.3.2 Turbulent Shear Stress The fundamental difference between laminar and turbulent flow lies in the chaotic, random behavior of the various fluid parameters. Such variations occur in the three components of velocity, the pressure, the shear stress, the temperature, and any other variable that has a field description. Turbulent flow is characterized by random, three-dimensional vorticity 1i.e., fluid particle rotation or spin; see Section 6.1.32. As is indicated in Fig. 8.12, such flows can be described in terms of their mean values 1denoted with an overbar2 on which are superimposed the fluctuations 1denoted with a prime2. Thus, if is the x component of instantaneous velocity, then its time mean 1or time-average2 value, is (8.24) where the time interval, T, is considerably longer than the period of the longest fluctuations, but considerably shorter than any unsteadiness of the average velocity. This is illustrated in Fig. 8.12. The fluctuating part of the velocity, , is that time-varying portion that differs from the average value (8.25) Clearly, the time average of the fluctuations is zero, since

Valves control the flowrate by providing a means to adjust the overall system loss coefficient to the desired value. When the valve is closed, the value of is infinite and no fluid flows. Opening of the valve reduces producing the desired flowrate. Typical cross sections of various types of valves are shown in Fig. 8.32. Some valves 1such as the conventional globe valve2 are designed for general use, providing convenient control between the extremes of fully closed and fully open. Others 1such as a needle valve2 are designed to provide very fine control of the flowrate. The check valve provides a diode type operation that allows fluid to flow in one direction only. Loss coefficients for typical valves are given in Table 8.2. As with many system components, the head loss in valves is mainly a result of the dissipation of kinetic energy of a high-speed portion of the flow. This high speed, is illustrated in Fig. 8.33.

We consider the fluid element at time t as is shown in Fig. 8.7. It is a circular cylinder of fluid of length and radius r centered on the axis of a horizontal pipe of diameter D. Because the velocity is not uniform across the pipe, the initially flat ends of the cylinder of fluid at time t become distorted at time when the fluid element has moved to its new location along the pipe as shown in the figure. If the flow is fully developed and steady, the distortion on each end of the fluid element is the same, and no part of the fluid experiences any acceleration as it flows, as shown by the figure in the margin. The local acceleration is zero because the flow is steady, and the convective acceleration is zero because the flow is fully developed. Thus, every part of the fluid merely flows along its streamline parallel to the pipe walls with constant velocity, although neighboring particles have slightly different velocities. The velocity varies from one pathline to the next. This velocity variation, combined with the fluid viscosity, produces the shear stress. If gravitational effects are neglected, the pressure is constant across any vertical cross section of the pipe, although it varies along the pipe from one section to the next. Thus, if the pressure is at section 112, it is at section 122 where is the pressure drop between sections (1) and (2). We anticipate the fact that the pressure decreases in the direction of flow so that A shear stress, acts on the surface of the cylinder of fluid. This viscous stress is a function of the radius of the cylinder, As was done in fluid statics analysis 1Chapter 22, we isolate the cylinder of fluid as is shown in Fig. 8.8 and apply Newton's second law, In this case, even though the fluid is moving, it is not accelerating, so that Thus, fully developed horizontal pipe flow is merely a

We will not repeat the detailed steps used to obtain the laminar pipe flow from the Navier - Stokes equations 1see Section 6.9.32 but will indicate how the various assumptions used and steps applied in the derivation correlate with the analysis used in the previous section. General motion of an incompressible Newtonian fluid is governed by the continuity equation 1conservation of mass, Eq. 6.312 and the momentum equation 1Eq. 6.1272, which are rewritten here for convenience: (8.13) (8.14) For steady, fully developed flow in a pipe, the velocity contains only an axial component, which is a function of only the radial coordinate For such conditions, the left-hand side of the Eq. 8.14 is zero. This is equivalent to saying that the fluid experiences no acceleration as it flows along. The same constraint was used in the previous section when considering for the fluid cylinder. Thus, with the Navier-Stokes equations become (8.15) The flow is governed by a balance of pressure, weight, and viscous forces in the flow direction, similar to that shown in Fig. 8.10 and Eq. 8.10. If the flow were not fully developed 1as in an entrance region, for example2, it would not be possible to simplify the Navier -Stokes equations to that form given in Eq. 8.15 1the nonlinear term would not be zero2, and the solution would be very difficult to obtain. Because of the assumption that the continuity equation, Eq. 8.13, is automatically satisfied. This conservation of mass condition was also automatically satisfied by the incompressible flow assumption in the derivation in the previous section. The fluid flows across one section of the pipe at the same rate that it flows across any other section 1see Fig. 8.82. When it is written in terms of polar coordinates 1as was done in Section 6.9.32, the component of Eq. 8.15 along the pipe becomes (8.16) Since the flow is fully developed, and the right-hand side is a function of, at most, only r. The left-hand side is a function of, at most, only x. It was shown that this leads to the condition that the pressure gradient in the x direction is a constant— The same condition was used in the derivation of the previous section 1Eq. 8.32. It is seen from Eq. 8.16 that the effect of a nonhorizontal pipe enters into the Navier-Stokes equations in the same manner as was discussed in the previous section. The pressure gradient in the flow direction is coupled with the effect of the weight in that direction to produce an effective pressure gradient of The velocity profile is obtained by integration of Eq. 8.16. Since it is a second-order equation, two boundary conditions are needed—112 the fluid sticks to the pipe wall 1as was also done in Eq. 8.72 and 122 either of the equivalent forms that the velocity remains finite throughout the flow 1in particular at 2 or, because of symmetry, that at In the derivation of the previous section, only one boundary condition 1the no-slip condition at the wall2 was needed because the equation integrated was a first-order equation. The other condition was automatically built into the analysis because of the fact that and at The results obtained by either applying to a fluid cylinder 1Section 8.2.12 or solving the Navier -Stokes equations 1Section 6.9.32 are exactly the same. Similarly, the basic assumptions regarding the flow structure are the same. This should not be surprising because the two methods are based on the same principle—Newton's second law. One is restricted to fully developed laminar pipe flow from the beginning 1the drawing of the free-body diagram2, and the other starts with the general governing equations 1the Navier -Stokes equations2 with the appropriate restrictions concerning fully developed laminar flow applied as the solution process progresses

and momentum equations for the control volume shown in Fig. 8.28 and the energy equation applied between 122 and 132. We assume that the flow is uniform at sections 112, 122, and 132 and the pressure is constant across the left-hand side of the control volume The resulting three governing equations 1mass, momentum, and energy2 are and These can be rearranged to give the loss coefficient, as where we have used the fact that This result, plotted in Fig. 8.27, is in good agreement with experimental data. As with so many minor loss situations, it is not the viscous effects directly 1i.e., the wall shear stress2 that cause the loss. Rather, it is the dissipation of kinetic energy 1another type of viscous effect2 as the fluid decelerates inefficiently. The losses may be quite different if the contraction or expansion is gradual. Typical results for a conical diffuser with a given area ratio, are shown in Fig. 8.29. 1A diffuser is a device shaped to decelerate a fluid.2 Clearly the included angle of the diffuser, is a very important parameter. For very small angles, the diffuser is excessively long and most of the head loss is due to the wall shear stress as in fully developed flow. For moderate or large angles, the flow separates from the walls and the losses are due mainly to a dissipation of the kinetic energy of the jet leaving the smaller diameter pipe. In fact, for moderate or large values of u 1i.e., for the case

balance between pressure and viscous forces—the pressure difference acting on the end of the cylinder of area and the shear stress acting on the lateral surface of the cylinder of area This force balance can be written as which can be simplified to give (8.3) Equation 8.3 represents the basic balance in forces needed to drive each fluid particle along the pipe with constant velocity. Since neither are functions of the radial coordinate, r, it follows that must also be independent of r. That is, where C is a constant. At 1the centerline of the pipe2 there is no shear stress At 1the pipe wall2 the shear stress is a maximum, denoted the wall shear stress. Hence, and the shear stress distribution throughout the pipe is a linear function of the radial coordinate (8.4) as is indicated in Fig. 8.9. The linear dependence of on r is a result of the pressure force being proportional to 1the pressure acts on the end of the fluid cylinder; 2 and the shear force being proportional to r 1the shear stress acts on the lateral sides of the cylinder; area 2. If the viscosity were zero there would be no shear stress, and the pressure would be constant throughout the horizontal pipe As is seen from Eqs. 8.3 and 8.4, the pressure drop and wall shear stress are related by (8.5) A small shear stress can produce a large pressure difference if the pipe is relatively long Although we are discussing laminar flow, a closer consideration of the assumptions involved in the derivation of Eqs. 8.3, 8.4, and 8.5 reveals that these equations are valid for both laminar and turbulent flow. To carry the analysis further we must prescribe how the shear stress is related to the velocity. This is the critical step that separates the analysis of laminar from that of turbulent flow—from being able to solve for the laminar flow properties and not being able to solve for the turbulent flow properties without additional ad hoc assumptions. As is discussed in Section 8.3, the shear stress dependence for turbulent flow is very complex. However, for laminar flow of a

dependent on the roughness of the pipe wall, unlike laminar pipe flow which is independent of roughness. Small roughness elements 1scratches, rust, sand or dirt particles, etc.2 can easily disturb this viscous sublayer 1see Section 8.42, thereby affecting the entire flow. An alternate form for the shear stress for turbulent flow is given in terms of the eddy viscosity, where (8.27) This extension of laminar flow terminology was introduced by J. Boussinesq, a French scientist, in 1877. Although the concept of an eddy viscosity is intriguing, in practice it is not an easy parameter to use. Unlike the absolute viscosity, which is a known value for a given fluid, the eddy viscosity is a function of both the fluid and the flow conditions. That is, the eddy viscosity of water cannot be looked up in handbooks—its value changes from one turbulent flow condition to another and from one point in a turbulent flow to another. The inability to accurately determine the Reynolds stress, is equivalent to not knowing the eddy viscosity. Several semiempirical theories have been proposed 1Ref. 32 to determine approximate values of L. Prandtl 11875-19532, a German physicist and aerodynamicist, proposed that the turbulent process could be viewed as the random transport of bundles of fluid particles over a certain distance, the mixing length, from a region of one velocity to another region of a different velocity. By the use of some ad hoc assumptions and physical reasoning, it was concluded that the eddy viscosity was given by Thus, the turbulent shear stress is (8.28) The problem is thus shifted to that of determining the mixing length, Further considerations indicate that is not a constant throughout the flow field. Near a solid surface the turbulence is dependent on the distance from the surface. Thus, additional assumptions are made regarding how the mixing length varies throughout the flow. The net result is that as yet there is no general, all-encompassing, useful model that can accurately predict the shear stress throughout a general incompressible, viscous turbulent flow. Without such information it is impossible to integrate the force balance equation to obtain the turbulent velocity profile and other useful information, as was done for laminar flow. 8.3.3 Turbulent Velocity Profile Considerable information concerning turbulent velocity profiles has been obtained through the use of dimensional analysis, experimentation, numerical simulations, and semiempirical theoretical efforts. As is indicated in Fig. 8.15, fully developed turbulent flow in a pipe can be broken into three regions which are characterized by their distances from the wall: the viscous sublayer very near the pipe wall, the overlap region, and the outer turbulent layer throughout the center portion of the flow. Within the viscous sublayer the viscous shear stress is dominant compared with the turbulent 1or Reynolds2 stress, and the random, eddying nature of the flow is essentially absent. In the outer turbulent layer the Reynolds stress is dominant, and there is considerable mixing and randomness to the flow. The character of the flow within these two regions is entirely different. For example, within the viscous sublayer the fluid viscosity is an important parameter; the density is unimportant. In the outer layer the opposite is true. By a careful use of dimensional analysis arguments for the flow in each layer and by a matching of the results in the common overlap layer, it has been possible to obtain the following conclusions about the turbulent velocity profile in a smooth pipe 1Ref. 52. In the viscous sublayer the velocity profile can be written in dimensionless form as

factor, which is defined to be In this text we will use only the Darcy friction factor.2 Thus, the friction factor for laminar fully developed pipe flow is simply (8.19) as shown by the figure in the margin. By substituting the pressure drop in terms of the wall shear stress 1Eq. 8.52, we obtain an alternate expression for the friction factor as a dimensionless wall shear stress (8.20) Knowledge of the friction factor will allow us to obtain a variety of information regarding pipe flow. For turbulent flow the dependence of the friction factor on the Reynolds number is much more complex than that given by Eq. 8.19 for laminar flow. This is discussed in detail in Section 8.4. 8.2.4 Energy Considerations In the previous three sections we derived the basic laminar flow results from application of or dimensional analysis considerations. It is equally important to understand the implications of energy considerations of such flows. To this end we consider the energy equation for incompressible, steady flow between two locations as is given in Eq. 5.89 (8.21) Recall that the kinetic energy coefficients, and compensate for the fact that the velocity profile across the pipe is not uniform. For uniform velocity profiles, whereas for any nonuniform profile, The head loss term, accounts for any energy loss associated with the flow. This loss is a direct consequence of the viscous dissipation that occurs throughout the fluid in the pipe. For the ideal 1inviscid2 cases discussed in previous chapters, and the energy equation reduces to the familiar Bernoulli equation discussed in Chapter 3 1Eq. 3.72. Even though the velocity profile in viscous pipe flow is not uniform, for fully developed flow it does not change from section 112 to section 122 so that Thus, the kinetic energy is the same at any section and the energy equation becomes (8.22) The energy dissipated by the viscous forces within the fluid is supplied by the excess work done by the pressure and gravity forces as shown by the figure in the margin. A comparison of Eqs. 8.22 and 8.10 shows that the head loss is given by 1recall and which, by use of Eq. 8.4, can be rewritten in the form (8.23) It is the shear stress at the wall 1which is directly related to the viscosity and the shear stress throughout the fluid2 that is responsible for the head loss. A closer consideration of the assumptions involved in the derivation of Eq. 8.23 will show that it is valid for both laminar and turbulent flow

flow separation is less severe, but there still are viscous effects. These are accounted for by use of the nozzle discharge coefficient, where (8.39) with As with the orifice meter, the value of is a function of the diameter ratio, and the Reynolds number, Typical values obtained from experiments are shown in Fig. 8.43. Again, precise values of depend on the specific details of the nozzle design. Accepted standards have been adopted 1Ref. 242. Note that the nozzle meter is more efficient 1less energy dissipated2 than the orifice meter. The most precise and most expensive of the three obstruction-type flow meters is the Venturi meter shown in Fig. 8.44 [G. B. Venturi (1746-1822)]. Although the operating principle for this device is the same as for the orifice or nozzle meters, the geometry of the Venturi meter is designed to reduce head losses to a minimum. This is accomplished by providing a relatively streamlined contraction 1which eliminates separation ahead of the throat2 and a very gradual expansion downstream of the throat 1which eliminates separation in this decelerating portion of the device2. Most of the head loss that occurs in a well-designed Venturi meter is due to friction losses along the walls rather than losses associated with separated flows and the inefficient mixing motion that accompanies such flow

for fluid that travels through pipes 112 and 132. These can be combined to give This is a statement of the fact that fluid particles that travel through pipe 122 and particles that travel through pipe 132 all originate from common conditions at the junction 1or node, N2 of the pipes and all end up at the same final conditions. The flow in a relatively simple looking multiple pipe system may be more complex than it appears initially. The branching system termed the three-reservoir problem shown in Fig. 8.37 is such a system. Three reservoirs at known elevations are connected together with three pipes of known properties 1lengths, diameters, and roughnesses2. The problem is to determine the flowrates into or out of the reservoirs. If valve 112 were closed, the fluid would flow from reservoir B to C, and the flowrate could be easily calculated. Similar calculations could be carried out if valves 122 or 132 were closed with the others open. With all valves open, however, it is not necessarily obvious which direction the fluid flows. For the conditions indicated in Fig. 8.37, it is clear that fluid flows from reservoir A because the other two reservoir levels are lower. Whether the fluid flows into or out of reservoir B depends on the elevation of reservoirs B and C and the properties 1length, diameter, roughness2 of the three pipes. In general, the flow direction is not obvious, and the solution process must include the determination of this direction. This is illustrated in Example 8.14.

include the water pipes in our homes and the distribution system that delivers the water from the city well to the house. Numerous hoses and pipes carry hydraulic fluid or other fluids to various components of vehicles and machines. The air quality within our buildings is maintained at comfortable levels by the distribution of conditioned 1heated, cooled, humidifieddehumidified2 air through a maze of pipes and ducts. Although all of these systems are different, the fluid mechanics principles governing the fluid motions are common. The purpose of this chapter is to understand the basic processes involved in such flows. Some of the basic components of a typical pipe system are shown in Fig. 8.1. They include the pipes themselves 1perhaps of more than one diameter2, the various fittings used to connect the individual pipes to form the desired system, the flowrate control devices 1valves2, and the pumps or turbines that add energy to or remove energy from the fluid. Even the most simple pipe systems are actually quite complex when they are viewed in terms of rigorous analytical considerations. We will use an "exact" analysis of the simplest pipe flow topics 1such as laminar flow in long, straight, constant diameter pipes2 and dimensional analysis considerations combined with experimental results for the other pipe flow topics. Such an approach is not unusual in fluid mechanics investigations. When "real-world" effects are important 1such as viscous effects in pipe flows2, it is often difficult or "impossible" to use only theoretical methods to obtain the desired results. A judicious combination of experimental data with theoretical considerations and dimensional analysis often provides the desired results. The flow in pipes discussed in this chapter is an example of such an analysis.

indicated. In previous considerations involving inviscid flow, the Reynolds number is 1strictly speaking2 infinite 1because the viscosity is zero2, and the flow most surely would be turbulent. However, reasonable results were obtained by using the inviscid Bernoulli equation as the governing equation. The reason that such simplified inviscid analyses gave reasonable results is that viscous effects were not very important and the velocity used in the calculations was actually the timeaveraged velocity, indicated in Fig. 8.12. Calculation of the heat transfer, pressure drop, and many other parameters would not be possible without inclusion of the seemingly small, but very important, effects associated with the randomness of the flow. Consider flow in a pan of water placed on a stove. With the stove turned off, the fluid is stationary. The initial sloshing has died out because of viscous dissipation within the water. With the stove turned on, a temperature gradient in the vertical direction, is produced. The water temperature is greatest near the pan bottom and decreases toward the top of the fluid layer. If the temperature difference is very small, the water will remain stationary, even though the water density is smallest near the bottom of the pan because of the decrease in density with an increase in temperature. A further increase in the temperature gradient will cause a buoyancy-driven instability that results in fluid motion—the light, warm water rises to the top, and the heavy, cold water sinks to the bottom. This slow, regular "turning over" increases the heat transfer from the pan to the water and promotes mixing within the pan. As the temperature gradient increases still further, the fluid motion becomes more vigorous and eventually turns into a chaotic, random, turbulent flow with considerable mixing, vaporization (boiling) and greatly increased heat transfer rate. The flow has progressed from a stationary fluid, to laminar flow, and finally to turbulent, multi-phase (liquid and vapor) flow. Mixing processes and heat and mass transfer processes are considerably enhanced in turbulent flow compared to laminar flow. This is due to the macroscopic scale of the randomness in turbulent flow. We are all familiar with the "rolling," vigorous eddy type motion of the water in a pan being heated on the stove 1even if it is not heated to boiling2. Such finite-sized random mixing is very effective in transporting energy and mass throughout the flow field, thereby increasing the various rate processes involved. Laminar flow, on the other hand, can be thought of as very small but finite-sized fluid particles flowing smoothly in layers, one over another. The only randomness and mixing take place on the molecular scale and result in relatively small heat, mass, and momentum transfer rates. Without turbulence it would be virtually impossible to carry out life as we now know it. Mixing is one positive application of turbulence, as discussed above, but there are other situations where turbulent flow is desirable. To transfer the required heat between a solid and an adjacent fluid 1such as in the cooling coils of an air conditioner or a boiler of a power plant2 would require an enormously large heat exchanger if the flow were laminar. Similarly, the required mass transfer of a liquid state to a vapor state 1such as is needed in the evaporated cooling system associated with sweating2 would require very large surfaces if the fluid flowing past the surface were 0T0z, u, 400 Chapter 8 ■ Viscous Flow in Pipes V8.4 Stirring color into paint V8.5 Laminar and turbulent mixing u(t) _ u = time-averaged (or mean) value u' T t O tO + T u t F I G U R E 8.12 The time-averaged, and fluctuating, description of a parameter for turbulent flow. u, u, JWCL068_ch08_383-460.qxd 9/23/08 10:51 AM Page 400 laminar rather than turbulent. As shown in Chapter 9, turbulence can also aid in delaying flow separation

of the center sphere, normal to the disk, completes one circle. The volume of fluid that has passed through the meter can be obtained by counting the number of revolutions completed. Another quantity-measuring device that is used for gas flow measurements is the bellows meter as shown in Fig. 8.49. It contains a set of bellows that alternately fill and empty as a result of the pressure of the gas and the motion of a set of inlet and outlet valves. The common household natural gas meter is of this type. For each cycle [1a2 through 1d2] a known volume of gas passes through the meter. The nutating disk meter 1water meter2 is an example of extreme simplicity—one cleverly designed moving part. The bellows meter 1gas meter2, on the other hand, is relatively complex—it contains many moving, interconnected parts. This difference is dictated by the application involved. One measures a common, safe-to-handle, relatively high-pressure liquid, whereas the other measures a relatively dangerous, low-pressure gas. Each device does its intended job very well. There are numerous devices used to measure fluid flow, only a few of which have been discussed here. The reader is encouraged to review the literature to gain familiarity with other useful, clever devices 1Refs. 25, 262

of the density than it is of viscosity. Second, we have introduced two additional dimensionless parameters, the Reynolds number, and the relative roughness, which are not present in the laminar formulation because the two parameters and are not important in fully developed laminar pipe flow. As was done for laminar flow, the functional representation can be simplified by imposing the reasonable assumption that the pressure drop should be proportional to the pipe length. 1Such a step is not within the realm of dimensional analysis. It is merely a logical assumption supported by experiments.2 The only way that this can be true is if the dependence is factored out as As was discussed in Section 8.2.3, the quantity is termed the friction factor, f. Thus, for a horizontal pipe (8.33) where For laminar fully developed flow, the value of f is simply independent of For turbulent flow, the functional dependence of the friction factor on the Reynolds number and the relative roughness, is a rather complex one that cannot, as yet, be obtained from a theoretical analysis. The results are obtained from an exhaustive set of experiments and usually presented in terms of a curve-fitting formula or the equivalent graphical form. From Eq. 5.89 the energy equation for steady incompressible flow is where is the head loss between sections 112 and 122. With the assumption of a constant diameter so that horizontal pipe with fully developed flow this becomes which can be combined with Eq. 8.33 to give h (8.34) L major f / D V2 2g ¢p p1 p2 ghL, 1a1 a2 1z 2, 1 z2 V 2 1 V2 1D 2, 1 D2 hL p1 g a1 V1 2 2g z1 p2 g a2 V2 2 2g z2 hL f f1Re, eD2, f 64Re, eD. f f aRe, e Db ¢p f / D rV2 2 ¢pD1/rV2 22 ¢p 1 2rV2 / D f aRe, e Db /D er Re rVDm, eD, 8.4 Dimensional Analysis of Pipe Flow 411 F I G U R E 8.19 Flow in the viscous sublayer near rough and smooth walls. R = D/2 δs δs Viscous sublayer Velocity profile, u = u(y) y x ∋ ∋ Rough wall Smooth wall or The major head loss in pipe flow is given in terms of the friction factor. JWCL068_ch08_383-460.qxd 9/23/08 10:52 AM Page 411 Equation 8.34, called the Darcy-Weisbach equation, is valid for any fully developed, steady, incompressible pipe flow—whether the pipe is horizontal or on a hill. On the other hand, Eq. 8.33 is valid only for horizontal pipes. In general, with the energy equation gives Part of the pressure change is due to the elevation change and part is due to the head loss associated with frictional effects, which are given in terms of the friction factor, f. It is not easy to determine the functional dependence of the friction factor on the Reynolds number and relative roughness. Much of this information is a result of experiments conducted by J. Nikuradse in 1933 1Ref. 62 and amplified by many others since then. One difficulty lies in the determination of the roughness of the pipe. Nikuradse used artificially roughened pipes produced by gluing sand grains of known size onto pipe walls to produce pipes with sandpaper-type surfaces. The pressure drop needed to produce a desired flowrate was measured and the data were converted into the friction factor for the corresponding Reynolds number and relative roughness. The tests were repeated numerous times for a wide range of Re and to determine the dependence. In commercially available pipes the roughness is not as uniform and well defined as in the artificially roughened pipes used by Nikuradse. However, it is possible to obtain a measure of the effective relative roughness of typical pipes and thus to obtain the friction factor. Typical roughness values for various pipe surfaces are given in Table 8.1. Figure 8.20 shows the functional dependence of f on Re and and is called the Moody chart in honor of L. F. Moody, who, along with C. F. Colebrook, correlated the original data of Nikuradse in terms of the relative roughness of commercially available pipe materials. It should be noted that the values of do not necessarily correspond to the actual values obtained by a microscopic determination of the average height of the roughness of the surface. They do, however, provide the correct correlation for It is important to observe that the values of relative roughness given pertain to new, clean pipes. After considerable use, most pipes 1because of a buildup of corrosion or scale2 may have a relative roughness that is considerably larger 1perhaps by an order of magnitude2 than that given. As shown by the figure in the margin, very old pipes may have enough scale buildup to not only alter the value of but also to change their effective diameter by a considerable amount. The following characteristics are observed from the data of Fig. 8.20. For laminar flow, which is independent of relative roughness. For turbulent flows with very large Reynolds numbers, which, as shown by the figure in the margin, is independent of the Reynolds number. For such flows, commonly termed completely turbulent flow 1or wholly turbulent flow2, the laminar sublayer is so thin 1its thickness decreases with increasing Re2 that the surface roughness completely dominates the character of the flow near the wall. Hence, the pressure drop required is a

result of an inertia-dominated turbulent shear stress rather than the viscosity-dominated laminar shear stress normally found in the viscous sublayer. For flows with moderate values of Re, the friction factor is indeed dependent on both the Reynolds number and relative roughness— The gap in the figure for which no values of f are given 1the range2 is a result of the fact that the flow in this transition range may be laminar or turbulent 1or an unsteady mix of both2 depending on the specific circumstances involved. Note that even for smooth pipes the friction factor is not zero. That is, there is a head loss in any pipe, no matter how smooth the surface is made. This is a result of the no-slip boundary condition that requires any fluid to stick to any solid surface it flows over. There is always some microscopic surface roughness that produces the no-slip behavior 1and thus 2 on the molecular level, even when the roughness is considerably less than the viscous sublayer thickness. Such pipes are called hydraulically smooth. Various investigators have attempted to obtain an analytical expression for Note that the Moody chart covers an extremely wide range in flow parameters. The nonlaminar region covers more than four orders of magnitude in Reynolds number—from to Obviously, for a given pipe and fluid, typical values of the average velocity do not cover this range. However, because of the large variety in pipes 1D2, fluids and and velocities 1V2, such a wide range in Re is needed to accommodate nearly all applications of pipe flow. In many cases the particular pipe flow of interest is confined to a relatively small region of the Moody chart, and simple semiempirical expressions can be developed for those conditions. For example, a company that manufactures cast iron water pipes with diameters between 2 and 12 in. may use a simple equation valid for their conditions only. The Moody chart, on the other hand, is universally valid for all steady, fully developed, incompressible pipe flows.

return to its fully developed character [section 152] and continues with this profile until the next pipe system component is reached [section 162]. In many cases the pipe is long enough so that there is a considerable length of fully developed flow compared with the developing flow length and In other cases the distances between one component 1bend, tee, valve, etc.2 of the pipe system and the next component is so short that fully developed flow is never achieved. 8.1.3 Pressure and Shear Stress Fully developed steady flow in a constant diameter pipe may be driven by gravity andor pressure forces. For horizontal pipe flow, gravity has no effect except for a hydrostatic pressure variation across the pipe, that is usually negligible. It is the pressure difference, between one section of the horizontal pipe and another which forces the fluid through the pipe. Viscous effects provide the restraining force that exactly balances the pressure force, thereby allowing the fluid to flow through the pipe with no acceleration. If viscous effects were absent in such flows, the pressure would be constant throughout the pipe, except for the hydrostatic variation. In non-fully developed flow regions, such as the entrance region of a pipe, the fluid accelerates or decelerates as it flows 1the velocity profile changes from a uniform profile at the entrance of the pipe to its fully developed profile at the end of the entrance region2. Thus, in the entrance region there is a balance between pressure, viscous, and inertia 1acceleration2 forces. The result is a pressure distribution along the horizontal pipe as shown in Fig. 8.6. The magnitude of the pressure gradient, is larger in the entrance region than in the fully developed region, where it is a constant, The fact that there is a nonzero pressure gradient along the horizontal pipe is a result of viscous effects. As is discussed in Chapter 3, if the viscosity were zero, the pressure would not vary with x. The need for the pressure drop can be viewed from two different standpoints. In terms of a force balance, the pressure force is needed to overcome the viscous forces generated. In terms of an energy balance, the work done by the pressure force is needed to overcome the viscous dissipation of energy throughout the fluid. If the pipe is not horizontal, the pressure gradient along it is due in part to the component of weight in that direction. As is discussed in Section 8.2.1, this contribution due to the weight either enhances or retards the flow, depending on whether the flow is downhill or uphill. The nature of the pipe flow is strongly dependent on whether the flow is laminar or turbulent. This is a direct consequence of the differences in the nature of the shear stress in laminar and turbulent flows. As is discussed in some detail in Section 8.3.3, the shear stress in laminar flow is a direct result of momentum transfer among the randomly moving molecules 1a microscopic phenomenon2. The shear stress in turbulent flow is largely a result of momentum transfer among the randomly moving, finite-sized fluid particles 1a macroscopic phenomenon2. The net result is that the physical properties of the shear stress are quite different for laminar flow than for turbulent flow

shown in Fig. 8.292, the conical diffuser is, perhaps unexpectedly, less efficient than a sharp-edged expansion which has There is an optimum angle 1 for the case illustrated2 for which the loss coefficient is a minimum. The relatively small value of for the minimum results in a long diffuser and is an indication of the fact that it is difficult to efficiently decelerate a fluid. It must be noted that the conditions indicated in Fig. 8.29 represent typical results only. Flow through a diffuser is very complicated and may be strongly dependent on the area ratio specific details of the geometry, and the Reynolds number. The data are often presented in terms of a pressure recovery coefficient, which is the ratio of the static pressure rise across the diffuser to the inlet dynamic pressure. Considerable effort has gone into understanding this important topic 1Refs. 11, 122. Flow in a conical contraction 1a nozzle; reverse the flow direction shown in Fig. 8.292 is less complex than that in a conical expansion. Typical loss coefficients based on the downstream 1high-speed2 velocity can be quite small, ranging from for to for for example. It is relatively easy to accelerate a fluid efficiently. Bends in pipes produce a greater head loss than if the pipe were straight. The losses are due to the separated region of flow near the inside of the bend 1especially if the bend is sharp2 and the swirling secondary flow that occurs because of the imbalance of centripetal forces as a result of the curvature of the pipe centerline. These effects and the associated values of for large Reynolds number flows through a bend are shown in Fig. 8.30. The friction loss due to the axial length of the pipe bend must be calculated and added to that given by the loss coefficient of Fig. 8.30. For situations in which space is limited, a flow direction change is often accomplished by use of miter bends, as is shown in Fig. 8.31, rather than smooth bends. The considerable losses in such bends can be reduced by the use of carefully designed guide vanes that help direct the flow with less unwanted swirl and disturbances. Another important category of pipe system components is that of commercially available pipe fittings such as elbows, tees, reducers, valves, and filters. The values of for such components depend strongly on the shape of the component and only very weakly on the Reynolds number for typical large Re flows. Thus, the loss coefficient for a elbow depends on whether the pipe joints are threaded or flanged but is, within the accuracy of the data, fairly independent of the pipe diameter, flow rate, or fluid properties 1the Reynolds number effect2. Typical values of for such components are given in Table 8.2. These typical components are designed more for ease of manufacturing and costs than for reduction of the head losses that they produce. The flowrate from a faucet in a typical house is sufficient whether the value of for an elbow is the typical or it is reduced to by use of a more expensive long-radius, gradual bend K 1Fig. 8.302.

standard electrical engineering methods can be carried over to help solve fluid mechanics problems, others cannot. One of the simplest multiple pipe systems is that containing pipes in series, as is shown in Fig. 8.35a. Every fluid particle that passes through the system passes through each of the pipes. Thus, the flowrate 1but not the velocity2 is the same in each pipe, and the head loss from point A to point B is the sum of the head losses in each of the pipes. The governing equations can be written as follows: and where the subscripts refer to each of the pipes. In general, the friction factors will be different for each pipe because the Reynolds numbers and the relative roughnesses will be different. If the flowrate is given, it is a straightforward calculation to determine the head loss or pressure drop 1Type I problem2. If the pressure drop is given and the flowrate is to be calculated 1Type II problem2, an iteration scheme is needed. In this situation none of the friction factors, are known, so the calculations may involve more trial-and-error attempts than for corresponding single pipe systems. The same is true for problems in which the pipe diameter 1or diameters2 is to be determined 1Type III problems2. Another common multiple pipe system contains pipes in parallel, as is shown in Fig. 8.35b. In this system a fluid particle traveling from A to B may take any of the paths available, with the total flowrate equal to the sum of the flowrates in each pipe. However, by writing the energy equation between points A and B it is found that the head loss experienced by any fluid particle traveling between these locations is the same, independent of the path taken. Thus, the governing equations for parallel pipes are and Again, the method of solution of these equations depends on what information is given and what is to be calculated. Another type of multiple pipe system called a loop is shown in Fig. 8.36. In this case the flowrate through pipe 112 equals the sum of the flowrates through pipes 122 and 132, or As can be seen by writing the energy equation between the surfaces of each reservoir, the head loss for pipe 122 must equal that for pipe 132, even though the pipe sizes and flowrates may be different for each. That is, for a fluid particle traveling through pipes 112 and 122, while

that the value of C is relatively insensitive to the shape of the conduit. Unless the cross section is very "thin" in some sense, the value of C is not too different from its circular pipe value, Once the friction factor is obtained, the calculations for noncircular conduits are identical to those for round pipes. Calculations for fully developed turbulent flow in ducts of noncircular cross section are usually carried out by using the Moody chart data for round pipes with the diameter replaced by the hydraulic diameter and the Reynolds number based on the hydraulic diameter. Such calculations are usually accurate to within about 15%. If greater accuracy is needed, a more detailed analysis based on the specific geometry of interest is needed

In a Type III problem we specify the pressure drop and the flowrate and determine the diameter of the pipe needed. For example, what diameter of pipe is needed between the water heater and dishwasher if the pressure in the water heater is 60 psi 1determined by the city water system2 and the flowrate is to be not less than 2.0 galmin 1determined by the manufacturer2? Several examples of these types of problems follow

vIn a Type III problem we specify the pressure drop and the flowrate and determine the diameter of the pipe needed. For example, what diameter of pipe is needed between the water heater and dishwasher if the pressure in the water heater is 60 psi 1determined by the city water system2 and the flowrate is to be not less than 2.0 galmin 1determined by the manufacturer2? Several examples of these types of problems follow

where D and f are based on the pipe containing the component. The head loss of the pipe system is the same as that produced in a straight pipe whose length is equal to the pipes of the original system plus the sum of the additional equivalent lengths of all of the components of the system. Most pipe flow analyses, including those in this book, use the loss coefficient method rather than the equivalent length method to determine the minor losses. Many pipe systems contain various transition sections in which the pipe diameter changes from one size to another. Such changes may occur abruptly or rather smoothly through some type of area change section. Any change in flow area contributes losses that are not accounted for in the fully developed head loss calculation 1the friction factor2. The extreme cases involve flow into a pipe from a reservoir 1an entrance2 or out of a pipe into a reservoir 1an exit2. A fluid may flow from a reservoir into a pipe through any number of differently shaped entrance regions as are sketched in Fig. 8.22. Each geometry has an associated loss coefficient. A typical flow pattern for flow entering a pipe through a square-edged entrance is sketched in Fig. 8.23. As was discussed in Chapter 3, a vena contracta region may result because the fluid cannot turn a sharp right-angle corner. The flow is said to separate from the sharp corner. The maximum velocity at section 122 is greater than that in the pipe at section 132, and the pressure there is lower. If this high-speed fluid could slow down efficiently, the kinetic energy could be converted into pressure 1the Bernoulli effect2, and the ideal pressure distribution indicated in Fig. 8.23 would result. The head loss for the entrance would be essentially zero. Such is not the case. Although a fluid may be accelerated very efficiently, it is very difficult to slow down 1decelerate2 a fluid efficiently. Thus, the extra kinetic energy of the fluid at section 122 is partially lost because of viscous dissipation, so that the pressure does not return to the ideal value. An entrance head loss 1pressure drop2 is produced as is indicated in Fig. 8.23. The majority of this loss is due to inertia effects that are eventually dissipated by the shear stresses within the fluid. Only a small portion of the loss is due to the wall shear stress within the entrance region. The net effect is that the loss coefficient for a square-edged entrance is approximately One-half of a velocity head is lost as the fluid enters the pipe. If the pipe protrudes into the tank 1a reentrant entrance2 as is shown in Fig. 8.22a, the losses are even greater. An obvious way to reduce the entrance loss is to round the entrance region as is shown in Fig. 8.22c, thereby reducing or eliminating the vena contracta effect. Typical values for the loss coefficient for entrances with various amounts of rounding of the lip a

where is the distance measured from the wall, is the time-averaged x component of velocity, and is termed the friction velocity. Note that u* is not an actual velocity of the fluid—it is merely a quantity that has dimensions of velocity. As is indicated in Fig. 8.16, Eq. 8.29 1commonly called the law of the wall2 is valid very near the smooth wall, for Dimensional analysis arguments indicate that in the overlap region the velocity should vary as the logarithm of y. Thus, the following expression has been proposed: (8.30) where the constants 2.5 and 5.0 have been determined experimentally. As is indicated in Fig. 8.16, for regions not too close to the smooth wall, but not all the way out to the pipe center, Eq. 8.30 gives a reasonable correlation with the experimental data. Note that the horizontal scale is a logarithmic scale. This tends to exaggerate the size of the viscous sublayer relative to the remainder of the flow. As is shown in Example 8.4, the viscous sublayer is usually quite thin. Similar results can be obtained for turbulent flow past rough walls 1Ref. 172. A number of other correlations exist for the velocity profile in turbulent pipe flow. In the central region 1the outer turbulent layer2 the expression where is the centerline velocity, is often suggested as a good correlation with experimental data. Another often-used 1and relatively easy to use2 correlation is the empirical power-law velocity profile (8.31) In this representation, the value of n is a function of the Reynolds number, as is indicated in Fig. 8.17. The one-seventh power-law velocity profile is often used as a reasonable approximation for many practical flows. Typical turbulent velocity profiles based on this power-law representation are shown in Fig. 8.18. A closer examination of Eq. 8.31 shows that the power-law profile cannot be valid near the wall, since according to this equation the velocity gradient is infinite there. In addition, Eq. 8.31 cannot be precisely valid near the centerline because it does not give at However, it does provide a reasonable approximation to the measured velocity profiles across most of the pipe. Note from Fig. 8.18 that the turbulent profiles are much "flatter" than the laminar profile and that this flatness increases with Reynolds number 1i.e., with n2. Recall from Chapter 3 that

where is the distance measured from the wall, is the time-averaged x component of velocity, and is termed the friction velocity. Note that u* is not an actual velocity of the fluid—it is merely a quantity that has dimensions of velocity. As is indicated in Fig. 8.16, Eq. 8.29 1commonly called the law of the wall2 is valid very near the smooth wall, for Dimensional analysis arguments indicate that in the overlap region the velocity should vary as the logarithm of y. Thus, the following expression has been proposed: (8.30) where the constants 2.5 and 5.0 have been determined experimentally. As is indicated in Fig. 8.16, for regions not too close to the smooth wall, but not all the way out to the pipe center, Eq. 8.30 gives a reasonable correlation with the experimental data. Note that the horizontal scale is a logarithmic scale. This tends to exaggerate the size of the viscous sublayer relative to the remainder of the flow. As is shown in Example 8.4, the viscous sublayer is usually quite thin. Similar results can be obtained for turbulent flow past rough walls 1Ref. 172. A number of other correlations exist for the velocity profile in turbulent pipe flow. In the central region 1the outer turbulent layer2 the expression where is the centerline velocity, is often suggested as a good correlation with experimental data. Another often-used 1and relatively easy to use2 correlation is the empirical power-law velocity profile (8.31) In this representation, the value of n is a function of the Reynolds number, as is indicated in Fig. 8.17. The one-seventh power-law velocity profile is often used as a reasonable approximation for many practical flows. Typical turbulent velocity profiles based on this power-law representation are shown in Fig. 8.18. A closer examination of Eq. 8.31 shows that the power-law profile cannot be valid near the wall, since according to this equation the velocity gradient is infinite there. In addition, Eq. 8.31 cannot be precisely valid near the centerline because it does not give at However, it does provide a reasonable approximation to the measured velocity profiles across most of the pipe. Note from Fig. 8.18 that the turbulent profiles are much "flatter" than the laminar profile and that this flatness increases with Reynolds number 1i.e., with n2. Recall from Chapter 3 that reasonable approximate results are often obtained by using the inviscid Bernoulli equation and by assuming a fictitious uniform velocity profile. Since most flows are turbulent and turbulent flows tend to have nearly uniform velocity profiles, the usefulness of the Bernoulli equation and the uniform profile assumption is not unexpected. Of course, many properties of the flow cannot be accounted for without including viscous effects.

with the motion of fluid particles, As the molecules dart across a given plane 1plane A- A, for example2, the ones moving upward have come from an area of smaller average x component of velocity than the ones moving downward, which have come from an area of larger velocity. The momentum flux in the x direction across plane A- A gives rise to a drag 1to the left2 of the lower fluid on the upper fluid and an equal but opposite effect of the upper fluid on the lower fluid. The sluggish molecules moving upward across plane A- A must be accelerated by the fluid above this plane. The rate of change of momentum in this process produces 1on the macroscopic scale2 a shear force. Similarly, the more energetic molecules moving down across plane A- A must be slowed down by the fluid below that plane. This shear force is present only if there is a gradient in otherwise the average x component of velocity 1and momentum2 of the upward and downward molecules is exactly the same. In addition, there are attractive forces between molecules. By combining these effects we obtain the well-known Newton viscosity law: where on a molecular basis is related to the mass and speed 1temperature2 of the random motion of the molecules. Although the above random motion of the molecules is also present in turbulent flow, there is another factor that is generally more important. A simplistic way of thinking about turbulent flow is to consider it as consisting of a series of random, three-dimensional eddy type motions as is depicted 1in one dimension only2 in Fig. 8.14b. (See the photograph at the beginning of this chapter.) These eddies range in size from very small diameter 1on the order of the size of a fluid particle2 to fairly large diameter 1on the order of the size of the object or flow geometry considered2. They move about randomly, conveying mass with an average velocity This eddy structure greatly promotes mixing within the fluid. It also greatly increases the transport of x momentum across plane A- A. That is, finite particles of fluid 1not merely individual molecules as in laminar flow2 are randomly transported across this plane, resulting in a relatively large 1when compared with laminar flow2 shear force. These particles vary in size but are much larger than molecules

Fluid Mechanics 8

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