A water transmission or supply pipeline is not just an enclosed tube— it is an entire sys-tem that transports water, either by using gravity or with the aid of pumping, from its source to the general vicinity of the demand. It typically consists of pipes or channels with their associated control works, pumps, valves, and other components. A transmission
sys-(PA )1
Fx Fy
y
x
CV
(PA )2 V1
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FIGURE 2.3 Force and momentum fluxes at an elbow.
tem is usually composed of a single-series line, as opposed to a distribution system that often consists of a complex network of interconnected pipes.
As we have mentioned, there are many practical questions facing the designer of such a system. Do the pipes, reservoirs and pumps have a great enough hydraulic capacity? Can the flow be controlled to achieve the desired hydraulic conditions? Can the system be operated economically? Are the pipes and connections strong enough to withstand both unsteady and steady pressures?
Interestingly, different classes of models are used to answer them, depending on the nature of the flow and the approximations that are justified. More specifically, issues of hydraulic capacity are usually answered by projecting demands (water requirements) and analyzing the system under steady flow conditions. Here, one uses the best available esti-mates of future demands to size and select the primary pipes in the system. It is the hydraulic capacity of the system, largely determined by the effective diameter of the pipeline, that links the supply to the demand.
Questions about the operation and sizing of pumps and reservoirs are answered by considering the gradual variation of demand over relatively short periods, such as over an average day or a maximum day. In such cases, the acceleration of the fluid is often negli-gible and analysts use a quasi-steady approach: that is, they calculate forces and energy balances on the basis of steady flow, but the unsteady form is used for the continuity equa-tion so that flows can be accumulated and stored.
Finally, the issue of required strength, such as the pressure rating of pipes and fittings, is answered by considering transient conditions. Thus, the strength of a pipeline is deter-mined at least in part by the pressures generated by a rapid transition between flow states.
In this stage, short-term and rapid motions must be taken into account, because large forces and dangerous pressures can sometimes be generated. Here, forces are balanced with accelerations, mass flow rates with pressure changes. These transient conditions are discussed in more detail in section 2.8 and in chapter 10.
A large number of different flow conditions are encountered in pipeline systems. To facilitate analysis, these conditions are often classified according to several criteria. Flow classification can be based on channel geometry, material properties, dynamic consider-ations (both kinematic and kinetic), or some other characteristic feature of the flow. For example, on the basis of fluid type and channel geometry, the flow can be classified as open-channel, pressure, or gas flow. Probably the most important distinctions are based on the dynamics of flow (i.e., hydraulics). In this way, flow is classified as steady or unsteady, turbulent or laminar, uniform or nonuniform, compressible or incompressible, or single phase or multiphase. All these distinctions are vitally important to the analyst:
collectively, they determine which physical laws and material properties are dominant in any application.
Steady flow: A flow is said to be steady if conditions at a point do not change with time. Otherwise a flow is unsteady or transient. By this definition, all turbulent flows, and hence most flows of engineering importance, are technically unsteady. For this reason, a more restrictive definition is usually applied: A flow is considered steady if the temporal mean velocity does not change over brief periods. Although the assumption is not for-mally required, pipeline flows are usually considered to be steady; thus, transient condi-tions represent an ‘abnormal’, or nonequilibrium, transition from one steady-state flow to another. Unless otherwise stated, the initial conditions in transient problems are usually assumed to be steady.
Steady or equilibrium conditions in a pipe system imply a balance between the physi-cal laws. Equilibrium is typified by steady uniform flow in both open channels and closed conduits. In these applications, the rate of fluid inflow to each segment equals the rate of
outflow, the external forces acting on the flow are balanced by the changes in momentum, and the external work is compensated for by losses of mechanical energy. As a result, the fluid generally moves down an energy gradient, often visualized as flow in the direction of decreasing hydraulic grade-line elevations (e.g., Fig. 2.1).
Quasi-steady flow. When the flow becomes unsteady, the resulting model that must be used depends on how fast the changes occur. When the rate of change is particular-ly slow, typicalparticular-ly over a period of hours or days, the rate of the fluids acceleration is negligible. However, fluid will accumulate or be depleted at reservoirs, and rates of demand for water may slowly adjust. This allows the use of a quasi-steady or extend-ed-duration simulation model.
Compressible and Incompressible. If the density of the fluid is constant—both in time and throughout the flow field—a flow is said to be incompressible. Thus, is not a function of position or time in an incompressible flow. If changes in density are permitted or reguined the flow is compressible.
Surge. When the rate of change in flow is moderate, typically occurring over a period of minutes, a surge model is often used. In North America, the term surge indicates an analysis of unsteady flow conditions in pipelines when the following assumptions are made: the fluid is incompressible (thus, its density is constant) and the pipe walls are rigid and do not deform. These two assumptions imply that fluid velocities are not a function of position along a pipe of constant cross-section and the flow is uniform. In other words, no additional fluid is stored in a length of pipe as the pressure changes; because velocities are uniform, the rate at which fluid enters a pipe is always equal to the rate of discharge.
However, the acceleration of the fluid and its accumulation and depletion from reservoirs are accounted for in a surge model.
Waterhammer. When rapid unsteady flow occurs in a closed conduit system, the tran-sient condition is sometimes marked by a pinging or hammering noise, appropriately called waterhammer. However, it is common to refer to all rapidly changing flow condi-tions by this term, even if no audible shock waves are produced. In waterhammer models, it is usually assumed that the fluid is slightly compressible, and the pipe walls deform with changes in the internal pressure. Waterhammer waves propagate with a finite speed equal to the velocity of sound in the pipeline.
The speed at which a disturbance is assumed to propagate is the primary distinction between a surge and a waterhammer model. Because the wavespeed parameter a is relat-ed to fluid storage, the wavesperelat-ed is infinite in surge or quasi-steady models. Thus, in effect, disturbances are assumed to propagate instantly throughout the pipeline system. Of course they do no such thing, because the wavespeed is a finite physical property of a pipe system, much like its diameter, wall thickness, or pipeline material. The implication of using the surge or quasi-steady approximation is that the unsteady behavior of the pipe system is controlled or limited by the rate at which the hydraulic boundary conditions (e.g., pumps, valves, reservoirs) at the ends of the pipe respond to the flow and that the time required for the pipeline itself to react is negligible by comparison.
Although unsteady or transient analysis is invariably more involved than is steady-state modeling, neglecting these effects in a pipeline can be troublesome for one of two reasons: the pipeline may not perform as expected, possibly causing large remedial expenses, or the line may be overdesigned with respect to transient conditions, possibly causing unnecessarily large capital costs. Thus, it is essential for engineers to have a clear physical grasp of transient behavior and an ability to use the computer’s power to maxi-mum advantage.
One interesting point is that as long as one is prepared to assume the flow is com-pressible, the importance of compressibility does not need to be known a priory. In fact,
all the incompressible, quasi-steady, and steady equations are special cases of the full tran-sient equations. Thus, if the importance of compressibility or acceleration effects is unknown, the simulation can correctly assume compressible flow behavior and allow the analysis to verify or contradict this assumption.
Redistribution of water, whatever model or physical devices are used, requires control of the fluid and its forces, and control requires an understanding not only of physical law but also of material properties and their implications. Thus, an attempt to be more specif-ic and quantitative about these matters will be made as this chapter progresses.
In steady flow, the fluid generally moves in the direction of decreasing hydraulic grade-line elevations. Specific devices, such as valves and transitions, cause local pressure drops and dissipate mechanical energy; operating pumps do work on the fluid and increase downstream pressures while friction creates head losses more or less uniformly along the pipe length. Be warned, however—in transient applications, this orderly situation rarely exists. Instead, large and sudden variations of both discharge and pressure can occur and propagate in the system, greatly complicating analysis.