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The cross-sections of closed conduits can be of any shape or size and can be made of a variety of materials. Engineering applications of the principles of ow in closed conduits include the design of municipal water-supply systems and transmission lines. The basic equations governing the ow of uids in closed conduits are the continuity, momentum, and energy equations. The most useful forms of these equations for application to pipe ow problems are derived in this chapter. The governing equations are presented in forms that are applicable to any uid owing in a closed conduit, but particular attention is given to the ow of water. The computation of ows in pipe networks is a natural extension of the ows in single pipelines, and methods of calculating ows and pressure distributions in pipeline systems are also described here.
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Chin pdf Water-Resources Engineering: International Edition, David A. Chin epub Water-Resources Engineering: International Edition, pdf David A. International Edition, the book Water-Resources Engineering: DRAFT as of August 25, 17 average ow velocity V , and the friction factor f that characterizes the shear stress of the uid on the pipe.
Some references name Equation 2. The occurrence and dierences between laminar and turbulent ow was later quantied by Osbourne Reynolds. The pipe behaves like a smooth pipe when the friction factor does not depend on the height of the roughness projections on the wall of the pipe and therefore depends only on the Reynolds number.
The smooth pipe case generally occurs at lower Reynolds numbers, when the roughness projections are submerged within the viscous boundary layer. At higher values of the Reynolds number, the thickness of the viscous boundary layer decreases and eventually the roughness projections protrude suciently far outside the viscous boundary layer that the shear stress of the pipe boundary is dominated by the hydrodynamic drag associated with the roughness projections into the main body of the ow.
Under these circumstances, the ow in the pipe becomes fully turbulent, the friction factor is independent of the Reynolds number, and the pipe is considered to be hydraulically rough. The ow is actually turbulent under both smooth-pipe and rough-pipe conditions, but the ow is termed fully turbulent when the friction factor is independent of the Reynolds number.
Between the smooth- and rough-pipe conditions, there is a transition region in which the friction factor depends on both the Reynolds number and the relative roughness. Colebrook developed the following relationship that asymptotes to the Prandl and von K arman relations 1.
Equation 2. Commercial pipes dier from Nikuradses experimental pipes in that the heights of the roughness projections are not uniform and are not uniformly distributed. In commercial pipes, an equivalent sand roughness, k s , is dened as the diameter of Nikuradses sand grains that would cause the same head loss as in the commercial pipe. The equivalent sand roughness, k s , of several commercial pipe materials are given in Table 2. These values of k s apply to clean new pipe only; pipe that has been.
Osbourne Reynolds to Coated 0. General 0. Cast iron 0. Lined with bitumen 0. Haestad Methods, Inc. DRAFT as of August 25, 19 in service for a long time usually experiences corrosion or scale buildup that results in values of k s that are orders of magnitude larger than the values given in Table 2. The rate of increase of k s with time depends primarily on the quality of the water being transported, and the roughness coecients for older water mains are usually determined through eld testing AWWA, The expression for the friction factor derived by Colebrook Equation 2.
The Moody diagram indicates that for Re 2,, the ow is laminar 64 Re f 5 Rough turbulent zone Transitional zone 10 3 10 4 Reynolds number, Re 1 0. Moody Diagram Source: Moody, L. Beyond a Reynolds number of , the ow is turbulent and the friction. This type of diagram was originally suggested by Blasius in and Stanton in Stanton and Pannell, The Moody diagram is sometimes called the Stanton diagram Finnemore and Franzini The dashed line in Figure 2. The equation of this dashed line is given by Mott as 1.
Although the Colebrook equation Equation 2. This minor inconvenience was circumvented by Jain , who suggested the following explicit equation for the friction factor 1. The Jain equation Equation 2. Water from a treatment plant is pumped into a distribution system at a rate of 4.
The diameter of the pipe is mm and is made of ductile iron. Estimate the pressure m downstream of the treatment plant if the pipeline remains horizontal. Compare the friction factor estimated using the Colebrook equation to the friction factor estimated using the Jain equation.
After 20 years in operation, scale buildup is expected to cause the equivalent sand roughness of the pipe to increase by a factor of Determine the eect on the water pressure m downstream of the treatment plant. C, which is equal to 1. The Colebrook equation required that f be determined iteratively, but the explicit Jain approximation for f is given by 1.
After 20 years, the equivalent sand roughness, ks, of the pipe is 2. This is quite a signicant drop and shows why velocities of 9. The problem in Example 3. The approach is summarized as follows: Flowrate for a Given Head Loss. In many cases, the owrate through a pipe is not controlled but attains a level that matches the pressure drop available.
For example, the owrate through faucets in home plumbing is determined by the gage pressure in the water main, which is relatively insensitive to the ow through the faucet. A useful approach to this problem that utilizes the DRAFT as of August 25, 23 Colebrook equation has been suggested by Fay , where the rst step is to calculate Re. Swamee and Jain combine Equations 2. A mm diameter galvanized iron service pipe is connected to a water main in which the pressure is kPa gage.
If the length of the service pipe to a faucet is 40 m and the faucet is 1. C , the Swamee-Jain equation Equation 2. In many cases, an engineer must select a size of pipe to provide a given level of service. For example, the maximum owrate and maximum allowable pressure drop may be specied for a water delivery pipe, and the engineer is required to calculate the minimum diameter pipe that will satisfy these design constraints.
Solution of this problem necessarily requires an iterative procedure. Streeter and Wylie have suggested the following steps 1. Assume a value of f. Using the new f, repeat the procedure until the new f agrees with the old f to the rst two signicant digits. If the length of the service pipe is 35 m and the head loss in the pipe is not to exceed 50 m, calculate the minimum pipe diameter that can be used.
Step 1: Using the Colebrook equation Equation 2. Step 2: The required pipe diameter is therefore equal to 0. A commercially available pipe with the closest diameter larger than mm should be used. The iterative procedure demonstrated in the previous example converges fairly quickly, and does not pose any computational diculty.
This method is illustrated by repeating the previous example. If the length of the service pipe is 35 m, and the head loss in the pipe is not to exceed 50 m, use the Swamee-Jain equation to calculate the minimum pipe diameter that can be used. By convention, the heat added to a system and the work done by a system are positive quantities. The normal stresses on the inow and outow boundaries of the control volume are equal to the pressure, p, with shear stresses tangential to the boundaries of the control volume.
The work done by a uid in the control volume is typically separated into work done against external pressure forces, W p , plus work done against rotating surfaces, W s , commonly referred to as the shaft work. The rotating element is called a rotor in a gas or steam turbine, an impeller in a pump, and a runner in a hydraulic turbine. Since v n is equal to zero over the impervious boundaries in contact with the uid system, Equation 2.
The constants 1 and 2 are determined by the velocity prole across the ow boundaries, and these constants are called kinetic energy correction factors.
If the velocity is constant across a ow boundary, then it is clear from Equation 2. Many practitioners incorrectly refer to Equation 2. Fundamental dierences between the energy equation and the Bernoulli equa- tion are that the Bernoulli equation is derived from the momentum equation, which is independent of the energy equation, and the Bernoulli equation does not account for uid friction.
The total head measures the average energy per unit weight of the uid owing across a pipe cross-section. The energy equation, Equation 2. The practical application of Equation 2. Head Loss Along Pipe energy, h, is plotted relative to a dened datum, and the locus of these points is called the energy grade line. In most water-supply applications the velocity heads are negligible and the hydraulic grade line closely approximates the energy grade line.
Both the hydraulic grade line and the energy grade line are useful in visualizing the state of the uid as it ows along the pipe and are frequently used in assessing the performance of uid delivery systems.
Most uid delivery systems, for example, require that the uid pressure remain positive, in which case the hydraulic grade line must remain above the pipe. In circumstances where additional energy is required to maintain acceptable pressures in pipelines, a pump is installed along the pipeline to elevate the energy grade line by an amount h s , which also elevates the hydraulic grade line by the same amount. This condition is illustrated in Figure 2.
It should also be clear from Figure 2. Velocity Prole. The momentum and energy correction factors, and , depend on the cross- sectional velocity distribution. The velocity distribution given by Equation 2. This equation, however, is not applicable within the small region close to the centerline of the pipe and is also not applicable in the small region close to the pipe boundary.
The pipe boundary v must also be equal to zero, but Equation 2. Although the prole ts the data close to the centerline of the pipe, it does not give zero slope at the centerline. In most engineering applications, and are taken as unity see Problem 2. Head Losses in Transitions and Fittings.
The head losses in straight pipes of constant diameter are caused by friction between the moving uid and the pipe boundary and are estimated using the Darcy-Weisbach equation. The loss coecients for several ttings and transitions are shown in Figure 2. Head losses in transitions and ttings are also called local head losses or minor head losses.
Loss Coecients for Transitions and Fittings Source: Roberson, John A. DRAFT as of August 25, 33 these head losses are a signicant portion of the total head loss in a pipe. Detailed descriptions of local head losses in various valve geometries can be found in Mott , and additional data on local head losses in pipeline systems can be found in Brater and colleagues A pump is to be selected that will pump water from a well into a storage reservoir.
Find the head that must be added by the pump. The pipeline system is shown in Figure 2. Pipeline System that the minor loss coecient for each of the bends is equal to 0. Taking the elevation of the water surface in the well to be equal to 0 m, and proceeding from the well to the storage reservoir where the head is equal to 5 m , the energy equation Equation 2. At C, the kinematic viscosity, , is equal to 1. Minor losses are frequently neglected in the analysis of pipeline systems.
As a general rule, neglecting minor losses is justied when, on average, there is a length of 1, diameters between each minor loss Streeter et al. Head Losses in Noncircular Conduits. Most pipelines are of circular cross-section, but ow of water in noncircular conduits is commonly encountered in cases such as rectangular culverts owing full.
Characterizing a noncircular conduit by the hydraulic radius, R, is necessarily approximate since conduits of arbitrary cross-section cannot be described with a single parameter. Secondary currents that are generated across a noncircular conduit cross-section to redistribute the shears are another reason why noncircular conduits cannot be completely characterized by the hydraulic radius Liggett, Olson and Wright state that this approach can be used for rectangular conduits where the ratio of sides, called the aspect ratio, does not exceed about 8.
Potter and Wiggert state that aspect ratios must be less than 4: Water ows through a rectangular concrete culvert of width 2 m and depth 1 m. Assume that the culvert ows full.
The head loss can be calculated using Equation 2. Friction losses in pipelines should generally be calculated using the Darcy-Weisbach equation. However, a minor inconvenience in using the Darcy-Weisbach equation to relate the friction loss to the ow velocity results from the dependence of the friction factor on the ow velocity; therefore, the Darcy-Weisbach equation must be solved simultaneously with the Colebrook equation.
In modern engineering practice, computer hardware and software make this a very minor inconvenience. In earlier years, however, this was considered a real problem, and various empirical head-loss formulae were developed to relate the head loss directly to the ow velocity. The most commonly used empirical formulae are the Hazen-Williams formula and the Manning formula.
Values of C H for a variety of commonly used pipe materials are given in Table 2. Solving Equations 2. New, unlined 0. Welded and seamless 0. Velon and Johnson ; Wurbs and James The Hazen-Williams equation is applicable to the ow of water at Outside of these conditions, use of the Hazen-Williams equation is strongly discouraged.
To further support these quantitative limitations, Street and colleagues and Liou have shown that the Hazen-Williams coecient has a strong Reynolds number dependence, and is mostly applicable where the pipe is relatively smooth and in the early part of its transition to rough ow.
In spite of these cautionary notes, the Hazen-Williams formula is frequently used in the United States for the design of large water-supply pipes without regard to its limited range of applicability, a practice that can have very detrimental eects on pipe design and could potentially lead to litigation Bombardelli and Garca, Values of n for a variety of commonly used pipe materials are given in Table 2.
Estimate the head loss over m using: Compare your results. Substituting ks, D, and Re into the Colebrook equation yields the friction factor, f, where 1. The perfor- mance criteria of these systems are typically specied in terms of minimum ow rates and pressure heads that must be maintained at the specied points in the network. The procedure for analyzing a pipe network usually aims at nding the ow distribution within the network, with the pressure distribution being derived from the ow distribution using the energy equation.
A typical pipe network is illustrated in Figure 2. Typical Pipe Network cilities, outows from consumer withdrawals or res. All outows are assumed to occur at network junctions. The basic equations to be satised in pipe networks are the continuity and energy equations.
The continuity equation requires that, at each node in the network, the sum of the outows is equal to the sum of the inows. This requirement is expressed by the relation NP j.
The energy equation requires that the heads at each of the nodes in the pipe network be consistent with the head losses in the pipelines connecting the nodes. There are two principal methods of calculating the ows in pipe networks: In the nodal method, the energy equation is expressed in terms of the heads at the network nodes, while in the loop method the energy equation is expressed in terms of the ows in closed loops within the pipe network.
The energy equation stated in Equation 2. Application of the nodal method in practice is usually limited to relatively simple networks. The high-pressure ductile-iron pipeline shown in Figure 2. The pipeline characteristics are given in the following tables. Assume that the ows are fully turbulent in all pipes.
The equivalent sand roughness, ks, of ductile-iron pipe is 0. Equations 2. This problem has been solved by assuming that the ows in all pipes are fully turbulent.
This is generally not known for sure a priori, and therefore a complete solution would require repeating the calculations until the assumed friction factors are consistent with the calculated owrates. This requirement is expressed by the relation NP i. Solution of this system of ow equations is complicated by the fact that the equations are nonlinear, and numerical methods must be used to solve for the ow distribution in the pipe network. The Hardy Cross method Cross, is a simple technique for hand- solution of the loop system of equations governing ow in pipe networks.
This iterative method was developed before the advent of computers, and much more ecient algorithms are used for numerical computations. In spite of this limitation, the Hardy Cross method is presented here to illustrate the iterative solution of the loop equations in pipe networks. However, in working with pipe networks, it is required that the algebraic sum of the head losses in any loop of the network see Figure 2. We must therefore dene a positive ow direction such as clockwise , and count head losses as positive in pipes when the ow is in the positive direction and negative when the ow is opposite to the selected positive direction.
Under these circumstances, the sign of the head loss must be the same as the sign of the ow direction. Further, when the ow is in the positive direction, positive values of Q require a positive correction to the head loss; when the ow is in the negative direction, positive values in Q also require a positive correction to the calculated head loss.
To preserve the algebraic relation among head loss, ow direction, and ow error Q , Equation 2. On the basis of Equation 2. The approximation given by Equation 2. Solving Equation 2. The steps to be followed in using the Hardy Cross method to calculate the ow distribution in pipe networks are: Assume a reasonable distribution of ows in the pipe network.
This assumed ow distribution must satisfy continuity. For each loop, i, in the network, calculate the quantities r ij Q j [Q j [ n1 and nr ij [Q j [ n1 for each pipe in the loop. Calculate the ow correction, Q i , using Equation 2. Add the correction algebraically to the estimated ow in each pipe.
Values of r ij occur in both the numerator and denominator of Equation 2. Proceed to another circuit and repeat step 2. Repeat steps 2 and 3 until the corrections Q i are small.
The application of the Hardy Cross method is best demonstrated by an example. Compute the distribution of ows in the pipe network shown in Figure 2. The ows are taken as dimensionless for the sake of illustration. Flows in Pipe Network Solution. The rst step is to assume a distribution of ows in the pipe network that satises continuity. The assumed distribution 44 of ows is shown in Figure 2.
The ow correction for each loop is calculated using Equation 2. As the above example illustrates, complex pipe networks can generally be treated as a combi- nation of simple circuits or loops, with each balanced in turn until compatible ow conditions exist in all loops.
Typically, after the ows have been computed for all pipes in a network, the elevation of the hydraulic grade line and the pressure are computed for each junction node. These pressures are then assessed relative to acceptable operating pressures. In practice, analyses of complex pipe networks are usually done using commercial computer programs that solve the system of continuity and energy equations that govern the ows in pipe 46 networks.
These computer programs, such as EPANET Rossman, , generally use algorithms that are computationally more ecient than the Hardy Cross method, such as the linear theory method, the Newton-Raphson method, and the gradient algorithm Lansey and Mays, They can be clas- sied into two main categories: Positive displacement pumps deliver a xed quantity of uid with each revolution of the pump rotor, such as with a piston or cylinder, while rotodynamic pumps add energy to the uid by accelerating it through the action of a rotating impeller.
Rotodynamic pumps are far more common in engineering practice and will be the focus of this section. Three types of rotodynamic pumps commonly encountered are centrifugal pumps, axial-ow pumps, and mixed-ow pumps. In centrifugal pumps, the ow enters the pump chamber along the axis of the impeller and is discharged radially by centrifugal action, as illustrated in Figure 2.
In mixed-ow pumps, outows have both radial and axial components. Typical centrifugal and axial-ow pump installations are illustrated in Figure 2. Key components of the centrifugal pump are a foot valve installed in the suction pipe to prevent Centrifugal pump a b Strainer Foot valve G ate valve Check valve S h a f t Propeller blade G uide vane M otor Figure 2.
Finnemore and Franzini If the suction line is empty prior to starting the pump, then the suction line must be primed prior to startup. Unless the water is known to be very clean, a strainer should be installed at the inlet to the suction piping.
The pipe size of the suction line should never be smaller than the inlet connection on the pump; if a reducer is required, it should be of the eccentric type since concentric reducers place part of the supply pipe above the pump inlet where an air pocket could form.
The discharge line from the pump should contain a valve close to the pump to allow service or pump replacement. The pumps illustrated in Figure 2. In multistage pumps, two or more impellers are arranged in series in such a way that the discharge from one impeller enters the eye of the next impeller. If a pump has three 48 impellers in series, it is called a three-stage pump.
Multistage pumps are typically used when large pumping heads are required. The performance of a pump is measured by the head added by the pump and the pump eciency.
The head added by the pump, h p , is equal to the dierence between the total head on the discharge side of the pump and the total head on the suction side of the pump, and is sometimes referred to as the total dynamic head. According to the Buckingham pi theorem, this relationship can be expressed as a relation between three dimensionless groups as follows gh p. In most cases, the ow through the pump is fully turbulent and viscous forces are negligible relative to the inertial forces.
Under these circumstances, the viscosity of the uid is neglected and Equation 2. A class of pumps that have the same shape is called a homologous series, and the performance characteristics of a homologous series of pumps are described by curves such as those in Figure 2. Pumps are selected to meet specic design conditions and, since the eciency of a pump varies with the operating condition, it is usually desirable to select a pump that operates at or near the point of maximum eciency, indicated by the point P in Figure 2.
The point of maximum eciency of a pump is commonly called the best eciency point bep , and is sometimes called the nameplate or design point.
Maintaining operation near the bep will allow a pump to function for years with little maintenance, and as the operating point moves away from the bep, pump thrust and radial loads increase which increases the wear on the pump DRAFT as of August 25, 49 Efficiency curve h max h K 1 K 2 P v 2 D 2 gh p vD 3 Q Figure 2.
Performance Curves of a Homologous Series of Pumps bearings and shaft. At the best eciency point in Figure 2. The specic speed, n s , is also called the shape number Hwang and Houghtalen, ; Wurbs and James, or the type number Douglas et al. The most ecient operating point for a homologous series of pumps is therefore specied by the specic speed.
This nomenclature is somewhat unfortunate since the specic speed is dimensionless and hence does not have units of speed. Although N s has dimensions, the units are seldom stated in practice. The required pump operating point gives the owrate, Q, and head, h p , required from the pump; the rotational speed, , is determined by the synchronous speeds of available motors; and the specic speed calculated from the required operating point is the basis for selecting the appropriate pump.
Since the specic speed is independent of the size of a pump, and all homologous pumps of varying sizes have the same specic speed, then the calculated specic speed at the desired operating point indicates the type of pump that must be selected to ensure optimal eciency.
The specic speeds in parentheses correspond to Ns given by Equation 2.
The types of pump that give the maximum eciency for given specic speeds, n s , are listed in Table 2. Table 2. This indicates that centrifugal pumps are most ecient at delivering low ows at high heads, while axial ow pumps are most ecient at delivering high ows at low heads. The eciencies of radial-ow centrifugal pumps increase with increasing specic speed, while the eciencies of mixed-ow and axial-ow pumps decrease with increasing specic speed.
Pumps with specic speed less than 0. Since axial-ow pumps are most ecient at delivering high ows at low heads, this type of pump is commonly used to move large volumes of water through major canals, and an example of this application is shown in Figure 2. Figure 2. In these cases, it is recommended to choose a DRAFT as of August 25, 51 pump with a specic speed that is close to and greater than the required specic speed.
In rare cases, a new pump may be designed to meet the design conditions exactly, however, this is usually very costly and only justied for very large pumps. An anity law for the power delivered to the uid, P, can be derived from the anity relations given in Equation 2.
The eect of changes in owrate on eciency can be estimated using the relation 0. If the motor is changed to 2, rpm, estimate the new performance curve.
The performance characteristics of a homologous series of pumps is given by ghp. For each model series and rotational speed, pump manufacturers provide a performance curve or characteristic curve that shows the relationship between the head, h p , added by the pump and the owrate, Q, through the pump. Pump Performance Curve Source: Goulds Pumps www.
In this case, the homologous series of pumps Model have impeller diameters ranging from In Figure 2. Also shown in Figure 2. This power input to the pump shaft is called the brake horsepower. The goal in pump selection is to select a pump that operates at a point of maximum eciency and with a net positive suction head that exceeds the minimum allowable value.
Pumps are placed in pipeline systems such as that illustrated in Figure 2. Pipeline System and Q for the pipeline system, and this relationship is commonly called the system curve.
Because the owrate and head added by the pump must satisfy both the system curve and the pump charac- teristic curve, Q and h p are determined by simultaneous solution of Equation 2. The resulting values of Q and h p identify the operating point of the pump. The location of the operating point on the performance curve is illustrated in Figure 2.
Operating Point in Pipeline System Example 2. Water is being pumped from a lower to an upper reservoir through a pipeline system similar to the one shown in Figure 2.
The reservoirs dier in elevation by Using this pump, what ow do you expect in the pipeline? If the calculated operating point of the pump coincides with the point of maximum eciency, which is the goal of pump selection, calculate the specic speed of the pump in U. Customary units and verify that a centrifugal-type pump should be used.
However, if the ow is fully turbulent, then the friction factor depends only on the relative roughness according to Equation 2. This assumption can now be veried by recalculating the friction factor with the Reynolds number corresponding to the calculated owrate.
After calculating the revised ow, the friction factor must again be calculated to see if it is equal to the assumed value. If not, the process is repeated until the assumed and calculated friction factors are equal. This process of vaporization is called cavitation. Cavitation is usually a transient phenomenon that occurs as water enters the low-pressure suction side of a pump and experiences the even lower pressures adjacent to the rotating pump impeller. As the water containing vapor cavities moves toward the high-pressure environment of the discharge side of the pump, the vapor cavities are compressed and ultimately implode, creating small localized high-velocity jets that can cause considerable damage to the pump machinery.
The damage caused by collapsing vapor cavities usually manifests itself as pitting of the metal casing and impeller, reduced pump eciency, and excessive vibration of the pump. The noise generated by imploding vapor cavities resembles the sound of gravel going through a centrifugal pump.
Since the saturation vapor pressure increases with temperature, a system that operates satisfactorily without cavitation during the winter may have problems with cavitation during the summer.
In applying either Equation 2. Absolute pressures are usually more convenient, since the vapor pressure is typically given as an absolute pressure.
A typical illustration of this is shown in Figure 2. In the case shown, the allowable net positive suction head ranges from 16 ft 4. The length of the pipeline between the reservoir and the suction side of the pump is 3. Calculate the available net positive suction head at the pump. If the pump manufacturer gives the required net positive suction head under the current operating conditions as 1.
C, and the saturated vapor pressure of water, pv, is 2. The head loss, hL, must be estimated using the Darcy-Weisbach equation and minor loss coecients. The owrate, Q, is C the kinematic viscosity, , is 1.
It is interesting to note that, although the practical limit for the suction lift is typically on the order of 7 m, diculties in keeping air out of the suction pipe frequently limit the suction lift to around 3 m Kay, Combinations of pumps are referred to as pump systems, and the pumps within these systems are typically arranged either in series or parallel.
The characteristic curve of a pump system is determined by the arrangement of pumps within the system. Consider the case of two identical pumps in series, illustrated in Figure 2. The ow through each pump is equal to Q, and the head added by each pump is h p. For the two-pump system, the ow through the system is equal to Q and the head added by the system is 2h p. Consequently, the characteristic curve of the two-pump in series system is related to the characteristic curve of each pump in that for any ow Q the head added by the system is twice the head added by a single pump, and the relationship between the single-pump characteristic curve and the two-pump characteristic curve is illustrated in Figure 2.
Pumps in Parallel characteristic curve by multiplying the ordinate of the single-pump characteristic curve h p by n. Pumps in series are used in applications involving unusually high heads. The case of two identical pumps arranged in parallel is illustrated in Figure 2. In this case, the ow through each pump is Q and the head added is h p ; therefore, the ow through the two-pump system is equal to 2Q, while the head added is h p.
Consequently, the characteristic curve of the two-pump system is derived from the characteristic curve of the individual pumps by multiplying the abscissa Q by two. This is illustrated in Figure 2. In a similar manner, the characteristic curves of systems containing n identical pumps in parallel can be derived from the single-pump characteristic curve by multiplying the abscissa Q by n. Pumps in parallel are used in cases where the desired owrate is beyond the range of a single pump and also to provide exibility in pump operations, since some pumps in the system can be shut down during low-demand conditions or for service.
This arrangement is common in sewage pump stations and water-distribution systems, where owrates vary signicantly during the course of a day. When pumps are placed either in series or parallel, it is usually desirable that these pumps be identical; otherwise, the pumps will be loaded unequally and the eciency of the pump system will be less than optimal.
In cases where nonidentical pumps are placed in series, the characteristic curve of the pump system is obtained by summing the heads added by the individual pumps for a given owrate. In cases where nonidentical pumps are placed in parallel, the characteristic curve of the pump system is obtained by summing the owrates through the individual pumps for a given head.
The major components of a water distribution system are pipelines, pumps, storage facilities, valves, and meters. The primary requirements of distribution system are to supply each customer with a sucient volume of water at adequate pressure, to deliver safe water that satises the quality expectations of customers, and to have sucient capacity and reserve storage for re protection.
AWWA, c. There are usually several categories of water demand within any populated area, and these sources of demand can be broadly grouped into residential, commercial, industrial, and public. Residential water use is associated with houses and apartments where people live; commercial water use is associated with retail businesses, oces, hotels, and restaurants; industrial water use is associated with manufacturing and processing operations; and public water use includes governmental facilities that use water.
Large industrial requirements are typically satised by sources other than the public water supply. A typical distribution of water use for an average city in the United States is given in Table 2. These rates vary from city to city as a result of dierences in local conditions that are unrelated DRAFT as of August 25, 61 Table 2. Solley, to the eciency of water use.
Water consumption is frequently stated in terms of the average amount of water delivered per day to all categories of water use divided by the population served, which is called the average per capita demand. The distribution of average per capita rates among water-supply systems serving approximately 95 million people in the United States is shown in Table 2.
Reprinted from Water Utility Operating Data, by permission. Copyright c American Wa- ter Works Association. Generally, high per capita rates are found in water-supply systems servicing large industrial or commercial sectors Dziegielewski et al. In the planning of municipal water-supply projects, the water demand at the end of the design life of the project is usually the basis for design.
For existing water-supply systems, the American Water Works Association AWWA, recommends that every 5 or 10 years, as a minimum, water-distribution systems be thoroughly reevaluated for requirements that would be placed on it by development and reconstruction over a year period into the future. The estimation of the design owrates for components of the water-supply system typically requires prediction of the population of the service area at the end of the design life, which is then multiplied by the per capita water demand to yield the design owrate.
Whereas the per capita water demand can usually be assumed to be fairly constant, the estimation of the future population typically involves a nonlinear extrapolation of past population trends. A variety of methods are used in population forecasting.
The simplest models treat the popula- 62 tion as a whole, forecast future populations based on past trends, and t empirical growth functions to historical population data. The most complex models disaggregate the population into various groups and forecast the growth of each group separately. A popular approach that segregates the population by age and gender is cohort analysis Sykes, High levels of disaggregation have the advantage of making forecast assumptions very explicit, but these models tend to be complex and require more data than the empirical models that treat the population as a whole.
Over rel- atively short time horizons, on the order of 10 years, detailed disaggregation models may not be any more accurate than using empirical extrapolation models of the population as a whole.
Several conventional extrapolation models are illustrated in the following paragraphs. Populated areas tend to grow in at varying rates, as illustrated in Figure 2. Growth Phases in Populated Areas stages of growth, there are wide open spaces. Integrating Equation 2. This phase of growth is called the declining growth phase. Almost all communities have zoning regulations that control the use of both developed and undeveloped areas within their jurisdiction sometimes called a master plan , and a review of these regulations will yield an estimate of the saturation population of the undeveloped areas.
The time scale associated with each growth phase is typically on the order of 10 years, although the actual duration of each phase can deviate signicantly from this number. The duration of each phase is important in that population extrapolation using a single-phase equation can only be justied for the duration of that growth phase. Consequently, single-phase extrapolations are typi- cally limited to 10 years or less, and these population predictions are termed short-term projections Viessman and Welty, Extrapolation beyond 10 years, called long-term projections, involve tting an S-shaped curve to the historical population trends and then extrapolating using the tted equation.
The conventional methodology to t the population equations to historical data is to plot the historical data, observe the trend in the data, and t the curve that best matches the population trend. You are in the process of designing a water-supply system for a town, and the design life of your system is to end in the year The population in the town has been measured every 10 years since by the U.
Census Bureau, and the reported populations are tabulated here. Estimate the population in the town using a graphical extension, b arithmetic growth projection, c geometric growth projection, d declining growth projection assuming a saturation concentration of , people , and e logistic curve projection. Year Population , , , , , , , , Solution. The population trend is plotted in Figure 2. Graphical extension to the year leads to a population estimate of , people.
Population Trend b Arithmetic growth is described by Equation 2. Consider the arithmetic projection of a line passing through points B and C on the approximate growth curve shown in Figure 2. Applying these conditions to Equation 2. Using points A and C in Figure 2. The projected results are compared graphically in Figure 2. Closer inspection of the predictions indicate that the declining and logistic growth models are limited by the specied saturation population of ,, while the geometric growth model is not limited by saturation conditions and produces the highest projected population.
The multiplication of the population projection by the per capita water demand is used to estimate the average daily demand for a municipal water-supply system. Variations in Demand. Water demand generally uctuates between being below the average daily demand in the early morning hours and above the average daily demand during the midday hours.
Typical daily cycles in water demand are shown in Figure 2. On a typical day in most 0 50 12 a. Noon 4 p. M idnight Figure 2. Linsley, Ray K. Water use rises rapidly in the morning 5 a. Use then increases in the evening 6 p. The range of demand conditions that are to be expected in water-distribution systems are specied by demand factors or peaking factors that express the ratio of the demand under certain 66 conditions to the average daily demand.
Typical demand factors for various conditions are given in Table 2. The demand factors in Table 2. Velon and Johnson In small water systems, demand factors may be signicantly higher than those shown in Table 2. Fire Demand. Besides the uctuations in demand that occur under normal operating conditions, water-distribution systems are usually designed to accommodate the large short-term water de- mands associated with ghting res.
Although there is no legal requirement that a governing body must size its water-distribution system to provide re protection, the governing bodies of most communities provide water for re protection for reasons that include protection of the tax base from destruction by re, preservation of jobs, preservation of human life, and reduction of human suering.
Flowrates required to ght res can signicantly exceed the maximum owrates in the absence of res, particularly in small water systems. In fact, for communities with populations less than 50,, the need for re protection is typically the determining factor in sizing water mains, storage facilities, and pumping facilities AWWA, c. In contrast to urban water systems, many rural water systems are designed to serve only domestic water needs, and re ow requirements are not considered in the design of these systems AWWA, c.
Numerous methods have been proposed for estimating re ows, the most popular of which was proposed by the Insurance Services Oce, Inc. ISO, , which is an organization representing the re insurance underwriters.
AWWA The maximum value of C i calculated using Equation 2. The occupancy factors, O i , for various classes of buildings are given in Table 2. Detailed tables for Table 2. Copyright c American Water Works Association. The NFF calculated using Equation 2. For one- and two-family dwellings not exceeding two stories in height, the NFF listed in Table 2. For other habitable buildings not listed in Table 2. Usually the local water utility will have a policy on the upper limit of re protection that it will provide to individual buildings.
Those wanting higher re ows need to either provide their own system or reduce re-ow requirements by installing sprinkler systems, re walls, or re-retardant materials Walski, ; AWWA, Estimates of the needed re ow calculated using Equation 2. The design duration of the re should follow the guidelines in Table 2. If these durations cannot be maintained, insurance rates are typically Table 2. Copyright c American Water Works Associ- ation. Estimate the owrate and volume of water required to provide adequate re protection to a story noncombustible building with an eective oor area of 8, m 2.
The NFF can be estimated by Equation 2. The occupancy factor, Oi, is given by Table 2. In high-value districts, additional hydrants may be necessary in the middle of long blocks to supply the required re ows. Fire hydrants may also be used to release air at high points in the water-distribution system and blow o sediments at low points in the system. Design Flows.
The design capacity of various components of the water-supply system are given in Table 2. The required capacities are based on per person use and population projections for full development of the service area. The required capac- ities shown in Table 2. Typically, the delivery pipelines from the water source to the treatment plant, as well as the treatment plant itself, are designed with a capacity equal to the maximum daily demand. However, because of the high cost of providing treatment, some utilities have trended towards using peak ows averaged over a longer period than one day to design water treatment plants such as 2 to 5 days , and relying on system storage to meet peak demands above treatment capacity.
This approach has serious water quality implications and should be avoided if possible AWWA, The owrates and pressures in the distribution system are analyzed under both maximum daily plus re demand and the maximum hourly demand, and the larger owrate governs the design. Pumps are sized for a variety of conditions from maximum daily to maximum hourly demand, depending on their function in the distribution system.
Additional reserve capacity is usually installed in water-supply systems to allow for redundancy and maintenance requirements. With a demand factor of 1. Source of supply: River indenite maximum daily demand Welleld maximum daily demand Reservoir average annual demand 2. Intake conduit maximum daily demand Conduit to treatment plant maximum daily demand 3. Low-lift 10 maximum daily demand, one reserve unit High-lift 10 maximum hourly demand, one reserve unit 4.
Treatment plant maximum daily demand 5. Service reservoir working storage plus re demand plus emergency storage 6. Distribution system: Supply pipe or conduit greater of 1 maximum daily demand plus re demand, or 2 maximum hourly demand Distribution grid full development same as for supply pipes Source: Reprinted by permission of Waveland Press, Inc.
All rights reserved. According to Table 2. The main supply pipe to the distribution system should therefore be designed with a capacity of 2. The water pressure within the distribution system must be above acceptable levels when the system demand is 2. Pipelines in water-distribution systems include transmission lines, arterial mains, and distribution mains. Transmission lines carry ow from the water-treatment plant to the service area, typically have diameters greater than mm, and are usually on the order of 3 km apart.
Arterial mains are connected to transmission mains and are laid out in interlocking loops with the pipelines not more than 1 km apart and diameters in the range of mm. Smaller form a grid over the entire service area, with diameters in the range of mm, and supply water to every user. Pipelines in distribution systems are collectively called water mains, and a pipe that carries water from a main to a building or property is called a service line.
Water mains are normally installed within the rights-of-way of streets. Dead ends in water-distribution systems should be avoided whenever possible, since the lack of ow in such lines may contribute to water-quality problems. Pipelines in water-distribution systems are typically designed with constraints relating to the minimum pipe size, maximum allowable velocity, and commercially available materials that will perform adequately under operating conditions. Minimum Size. The size of a water main determines its carrying capacity.
Main sizes must be selected to provide the capacity to meet peak domestic, commercial, and industrial demands in the area to be served, and must also provide for re ow at the necessary pressure. For re protection, insurance underwriters typically require a minimum main size of mm for residential areas and mm for high-value districts such as sports stadiums, shopping centers, and libraries if cross-connecting mains are not more than m apart.
On principal streets, and for all long lines not connected at frequent intervals, mm and larger mains are required. Service Lines. Service lines are pipes, including accessories, that carry water from the main to the point of service, which is normally a meter setting or curb stop located at the property line. Service lines can be any size, depending on how much water is required to serve a particular customer.
To properly size service lines it is essential to know the peak demands than any service tap will be called on to serve. A common method to estimate service ows is to sum the xture units associated with the number and type of xtures served by the service line and then use a curve called the Hunter curve to relate the peak owrate to total xture units. Recent research has indicated that the peak ows estimated from xture units and the Hunter curve provide conservative estimates of peak ows AWWA, Irrigation demands that occur simultaneously with peak domestic demands must be added to the estimated peak domestic demands.
Service lines are sized to provide an adequate service pressure downstream of the water meter when the service line is delivering the peak ow. This requires that the pressure and elevation at the tap, length of service pipe, head loss at the meter, elevation at the water meter, valve losses, and desired pressure downstream of the meter be known. Using the energy equation, the minimum service line diameter is calculated using this information.
It is usually better to overdesign a service line than to underdesign a service line because of 72 the cost of replacing a service line if service pressures turn out to be inadequate.
Materials used for service- line pipe and tubing are typically either copper tubing or plastic, which includes polyvinyl chloride PVC and polyethylene PE. Type K copper is the most commonly used material for copper service lines.
Older service lines used lead and galvanized iron, which are no longer recommended. The valve used to connect a small-diameter service line to a water main is called a corporation stop, which is sometimes loosely referred to as the corporation cock, corporation tap, corp stop, corporation, or simply corp or stop AWWA, c. Tapping a water main and inserting a corporation stop directly into the pipe wall requires a tapping machine, and taps are typically installed at the 10 or 2 oclock position on the pipe.
Good construction practices must be used when installing service lines to avoid costly repairs in the future. This must include burying the pipe below frost lines, maintaining proper ditch conditions, proper backll, trench compaction, and protection from underground structures that may cause damage to the ppe. Allowable Velocities. Maximum allowable velocities in pipeline systems are imposed to control friction losses and hydraulic transients.
Maximum allowable velocities of 0. The importance of controlling the maximum velocities in water distribution systems is supported by the fact that a change in velocity of 0. Pipeline materials should generally be selected based on a consideration of service con- ditions, availability, properties of the pipe, and economics.
In selecting pipe materials the following considerations should be taken into account: CIP is no longer manufactured in the United States. For new distribution mains, ductile iron pipe DIP is most widely used for pipe diam- eters up to mm 30 in. DIP is manufac- tured in diameters from 76 to mm in. For diameters from to mm in. The standard lengths of DIP are 5. DIP is usually coated outside and inside with an bituminous coating to minimize corrosion. An internal cement-mortar lining 1.
DIP used in water systems in the United States are provided with a cement-mortar lining unless otherwise specied by the downloadr. A variety of joints are available for use with DIP, which includes push-on the most common , mechanical, anged, ball-and-socket, and numerous joint designs. A stack of DIP is shown in Figure 2. A rubber gasket, to ensure a tight t, is contained in the bell side of the pipe. As a consequence, steel pipe is primarily used for transmission lines in water distri- bution systems.
Steel pipe available in diameters from to mm in. The standard length of steel pipe is The interior of steel pipe is usually pro- tected with either cement mortar or epoxy, and the exterior is protected by a variety of plastic coatings, bituminous materials, and polyethylene tapes depending on the degree of protection required. PVC pipe is by far the most widely used type of plastic pipe material for small-diameter water mains. PVC pipe is commonly available in diameters from to mm.
Extruded PE and PB pipe are primarily used for water service pipe in small sizes, however, the use of PB has decreased remarkably because of structural diculties caused by premature pipe failures.
In the hydraulic design of PVC pipes, a roughness height of 0. Asbestos-cement A-C pipe has been widely installed in water distribution systems, especially in areas where metallic pipe is subject to corrosion, such as in coastal areas.
It has also been installed in remote areas where its light weight makes it much easier to install than CIP. Common diameters are in the range of to mm. The U. Environmental Protection Agency banned most uses of asbestos in and, due to the manufacturing ban, new A-C pipe is no longer being installed in the United States.
Fiberglass pipe is available for potable water used is sizes from 25 to mm. Ad- vantages of berglass pipe include corrosion resistance, light weight, low installation cost, ease of repair, and hydraulic smoothness. Disadvantages include susceptibility to mechanical damage, low modulus of elasticity, and lack of standard joining system.
The use of concrete pressure pipe has grown rapidly since The pipe provides a combination of the high tensile strength of steel and the high compressive strength and corrosion resistance of concrete. The pipe is available in diameters ranging from to mm and in standard lengths from 3.
Concrete pipe is available with various types of liners and reinforcement, and the four types in common use in the United States and Canada are: Pipelines in water-distribution systems should be buried to a depth below the frost line in northern climates and at a depth sucient to cushion the pipe against trac loads in warmer climates Clark, Generally, a cover of 1.
In areas where frost penetration is a signicant factor, mains can have as much as 2. Trenches for water mains should be as narrow as possible and still be wide enough to allow for proper joining and compaction around the pipe. The suggested trench width is the nominal pipe diameter plus 0. Trench bottoms should be undercut 15 to 25 cm, and sand, clean ll, or crushed stone installed to provide a cushion against the bottom of the excavation, which is usually rock Clark, Standards for pipe construction, installation, and performance are published by the American Water Works Association in its C-series standards, which are continuously being updated.
When portions of the distribution system are separated by long distances or signicant changes in elevation, booster pumps are sometimes used to maintain acceptable service pressures. In some cases, re-service pumps are used to provide additional capacity for emergency re protection.
Pumps operate at the intersection of the pump performance curve and the system curve. Since the system curve is signicantly aected by variations in water demand, there is a signicant variation in pump operating conditions.
In most cases, the range of operating conditions is too wide to be met by a single pump, and multiple-pump installations or variable speed pumps are required Velon and Johnson, Shuto valves or gate valves are typically provided at m intervals so that areas within the system can be isolated for repair or maintenance; air-relief valves or air-and-vacuum relief valves are required at high points to release trapped air; blowo valves or drain valves may be required at low points; and backow prevention devices are required by applicable regulations to prevent contamination from backows of nonpotable water into the distribution system from system outlets.
To maintain the performance of water-distribution systems it is recommended that each valve should be operated through a full cycle and then returned to its normal position on a regular schedule. The time interval between operations should be determined by the manufacturers recommendations, size of the valve, severity of the operating conditions, and the importance of the installation AWWA, d. Unlike the service line and water tap, whic when incorrectly sized will generally require expensive excavation and retapping, water meters can usually be changed less expensively.
Selection of the type and size of a meter should be based primarily on the range of ow, and the pressure loss through the meter should also be a consideration.