The following variables and nomenclature are used throughout the liquid flow section:
Cw ⫽ Hazen-Williams Coefficient D ⫽ Inside pipe diameter, ft
d ⫽ Inside pipe diameter, in.
f ⫽ Friction factor, dimensionless g ⫽ Gravitational acceleration, ft/sec2 gc ⫽ Gravitational constant, 32.174 ft/sec2 hp ⫽ Head gain, ft
hL ⫽ Head loss, ft hf ⫽ Friction head loss, ft
hm ⫽ Head loss due to minor loss valve or fitting, ft hw ⫽ H2O pressure, in.
K ⫽ Resistance coefficient for valve or fitting k ⫽ Internal pipe wall roughness, ft
Lf ⫽ Pipe length, ft P ⫽ Pressure, lb/in2(psia) p ⫽ Pressure, lb/ft2, psf
⌬P ⫽ Change in pressure, psia P1 ⫽ Inlet or upstream pressure, psia P2 ⫽ Outlet or downstream pressure, psia Q ⫽ Flow rate, gallons/min
Qh ⫽ Volumetric flow rate, ft3/hr (cfh) Re ⫽ Reynolds number, dimensionless
⫽ Kinematic viscosity, ft2/sec
⫽ Absolute (dynamic) viscosity, lbm/ft-sec
⫽ Density of fluid, lb/ft3
␥ ⫽ Specific weight, lb/ft3
Table 3.1 lists some general formulas used for liquid hydraulics.
Table 3.2 lists some conversion factors used in liquid hydraulics.
The Energy Principle
Although there is no such thing as a truly incompressible fluid, this term is used for liquids. The first law of thermodynamics states that for any given system, the change in energy is equal to the difference between the heat transferred to the
sys-PLASTIC PIPING HANDBOOK 3.2
tem and the work done by the system on its surroundings during a given time interval. This energy represents the total energy of the system. In piping applica-tions, energy often is converted into units of energy per unit weight resulting in units of length. Engineers use these length equivalents to get a better feel for the resulting behavior of the system. In pipeline hydraulics, we express the state of the system in terms of “head” or feet of head. The energy at any point in a piping system often is identified as:
Pressure head ⫽ p/␥
Elevation head⫽ z Velocity head ⫽ v2/2g
FLUID FLOW 3.3
TABLE 3.1 General Formulas Used for Liquid Hydraulics
Formulas Symbols D⫽ Inside diameter of pipe, ft H⫽ Pressure measured in ft of head P⫽ Pressure measured in lb/in2 Q⫽ Flow rate in ft3/sec V⫽ Pipeline fill per ft
G⫽ Pipeline fill per length in gal B⫽ Pipeline fill per length in barrels
TABLE 3.2 Conversion Factors
1 ft3⫽ 7.48 gallons ft3/sec⫽ 642 BPH 1 barrel⫽ 42 gallons ft3/sec⫽ 449 GPM 1 gallon⫽ 231 in3 1 GPM⫽ 1.43 BPH 1 ft3⫽ 1728 in3
These quantities can be used to express the head loss or head gain between two locations using the energy equation.
The Energy Equation
In addition to pressure head, elevation head, and velocity head, head also can be added to the system (usually by a pump) and head can be removed from the sys-tem due to friction or other disturbances within the syssys-tem. These changes in head are referred to as head gains and head losses. By balancing the energy between two points in the system, we can obtain the energy equation (Bernoulli’s Equation):
⫹ z1⫹ ⫹ hp⫽ ⫹ z2⫹ ⫹ hL (3.1)
The basic approach to all piping systems is to write the Bernoulli Equation between two points, connected by a streamline, where the conditions are known.
The total head at point 1 must match with the total head at point 2, adjusted for any increases in head because of pumps, losses because of pipe friction, and so-called “minor losses” because of entries, exits, fittings, etc. The parts of the energy equation can be combined to express two useful quantities, the hydraulic grade and the energy grade.
Hydraulic and Energy Grades
The hydraulic grade line (HGL) and the energy grade line (EGL) are two useful engineering tools in the hydraulic design of a system that is in a dynamic state. The hydraulic grade is the sum of the pressure head and the elevation head. This rep-resents the height that a water column would raise in a piezometer. When plotted in a profile, this is referred to as the hydraulic grade line or HGL (see Figure 3.1).
The energy grade is the sum of the hydraulic grade and the velocity head and represents the height that a column of water would raise in a pitot tube. When plot-ted in a profile, this is referred to as the energy grade line, or EGL (see Figure 3.1).
Pipe Sizing
Fluid flow is a basic component of sizing a piping system. The fluid flow design determines the minimum acceptable pipe diameter required for transferring the fluid efficiently. The main factors in determining the minimum acceptable pipe diameter are the design flow rates and pressures losses. The design flow rates are based on system demands that usually are established in the design phase of a project. Before the determination of the minimum inside diameter can be made, service conditions must be reviewed to determine operational requirements, such as the recommended fluid velocity, and liquid characteristics, such as viscosity, temperature, and solids density.
For normal liquid service applications, the acceptable fluid velocity in pipes is around 7 ft/sec⫾ 3 ft/sec. The maximum velocity at piping discharge points usu-ally is limited to 7 ft/sec. These velocity ranges are considered reasonable design targets for normal applications. Other limiting factors, such as pressure transient conditions, however, can overrule. In addition, some applications can allow greater velocities based on general industry practices, such as boiler feed water and petroleum liquids.
Pressure losses throughout a piping system should be designed to provide an optimum balance between the installed cost of a piping system and operating cost of the system. The primary factors that will affect the cost and system operating performances are the inside pipe diameter (and the resulting fluid velocity), mate-rials of construction, and pipe routing.
Energy Losses in Pipes
When a fluid is transported inside a pipe, the pipe’s inside diameter determines the allowable flow rate. Several factors might cause the energy loss (hL) in a pip-ing system, with the main cause friction between the fluid and the pipe wall. Liq-uids in the pipe resist flowing because of viscous shear stresses within the fluid and friction along the pipe walls. This friction is present throughout the length of
FLUID FLOW 3.5
FIGURE 3.1 Energy grade line.
the pipe. As a result, the energy grade line (EGL) and the hydraulic grade line (HGL) drop linearly in the direction of flow. Flow resistance in pipe results in a pres-sure drop, or loss of head, in the piping system.
Localized areas of increased turbulence and disruption of the streamlines are secondary causes of energy loss. These disruptions usually are caused by valves, meters, or fittings and are referred to as minor losses. When considered against the friction losses within a piping system, the minor losses often are considered neg-ligible and sometimes are not considered in an analysis. While the term minor loss often is applicable for large piping systems, it might not always be the case. In pip-ing systems that have numerous valves and fittpip-ings relative to the total length of pipe, the minor losses can have a significant impact on the energy or head losses.
Pressure Flow of Liquids
Many equations approximate the friction losses that can be expected with the flow of liquid through a pressure pipe. The two most frequently used equations in plas-tic piping systems are:
Darcy-Weisbach Equation Hazen-Williams Equation
The Darcy-Weisbach Equation applies to a wide range of fluids, while the Hazen-Williams Equation is based on empirical data and is used primarily in water modeling applications. Each of these methods calculates friction losses as a function of the velocity of the fluid and some measure of the pipe’s resistance to flow (pipe wall roughness). Typical pipe roughness values for these methods are shown in Table 3.3. These values can vary depending on the product manu-facturer, workmanship, age, and many other factors.
Darcy-Weisbach Equation. Friction losses in a piping system are a complex function of the system geometry, the fluid properties, and the flow rate in the system. By observation, the head loss is roughly proportional to the square of the flow rate in most engineering flows (fully developed, turbulent pipe flow). This observation leads us to the Darcy-Weisbach Equation for head loss from friction:
hf⫽ f (3.2)
The Darcy-Weisbach Equation is a generally accepted method for calculating fric-tion losses from liquids flowing in full pipes. It recognizes the dependence on pipe diameter, pipe wall roughness, liquid viscosity, and flow velocity. Darcy-Weisbach is a general equation that applies equally well at any flow rate and any incompres-sible fluid.
Depending upon the Reynolds number, the friction factor is a function of the relative wall roughness of the pipe, the velocity of the fluid, and the kinematic
v2
viscosity of the fluid. Liquid flow in pipes can be laminar or turbulent, or it can be in a transition between the two. For laminar flow (Reynolds number below 2000), the head loss is proportional to the velocity rather than the velocity squared and the pipe wall roughness has no effect. The friction factor calculation is:
f⫽ (3.3)
Laminar flow can be characterized as consisting of a series of thin shells that are sliding over one another. The velocity of the fluid is the greatest at the center and the velocity at the pipe wall is zero.
In the turbulent flow region, it is not possible to obtain an analytical solution for the friction factor as we do for laminar flow. Most of the data available for evalu-ating the friction factor in turbulent flow have been derived from experiments. For turbulent flow (Reynolds number above 4000), the friction factor is dependent upon the pipe wall roughness as well as the Reynolds number. For turbulent flow, Colebrook (1939) found an implicit correlation for the friction factor in round pipes. This correlation converges well in a few iterations.
⫽ ⫺2 log
[
⫹]
(3.4)or
RE⫽ (3.6)
The familiar Moody Diagram is a log-log plot of the Colebrook correlation on an axis of the friction factor and the Reynolds number, combined with the f⫽ 64/Re result for laminar flow.
For turbulent flow, appropriate values for the friction factor can be deter-mined using the Swamme and Jain Equation, which provides values within 1 per-cent of the Colebrook Equation over most of the useful ranges:
(3.7)
Hazen-Williams Equation The Hazen-Williams Equation is used primarily in the design and analysis of pressure pipe for water distribution systems. This equa-tion was developed experimentally with water and, under most condiequa-tions, should not be used for other fluids. The Hazen-Williams formula for water at 60°F, how-ever, can be applied to liquids that have the same kinematic viscosity as water. This
3162Q
ᎏdk
PLASTIC PIPING HANDBOOK 3.8
f⫽
[
In冢ᎏ3.7dk ⫹ᎏ5.74RE0.9冣]
21.325
FIGURE 3.2 Reynolds number.
equation includes a roughness factor Cw, which is constant over a wide range of turbulent flows and an empirical constant.
hf⫽ 冢 冣1.85 (3.8)
For a simpler solution to fluid flow in plastic pipe, consider this version of the Hazen-Williams formula:
⌬P100⫽ (3.8)
where ⌬P ⫽ Friction pressure loss, psi, per 100 feet of pipe.
The coefficient Cw is essentially a friction factor. Table 3.1 lists Cw values for various types of pipe.
The designer must use proper judgment to select pipe sizes that best meet the project conditions. The following considerations may be helpful:
• At a given flow rate, a larger diameter pipe will have a lower velocity and less pressure drop.
• At a given flow rate, a smaller diameter pipe will have higher velocity and increased pressure drop.
• The frictional head loss is less in larger diameter pipes than smaller pipe flowing at same velocity.
Minor Losses. Fluids flowing through a valve or fitting will have a friction head loss. Minor losses in pipes at these areas are caused by increased turbulence, which causes a drop in the energy and hydraulic grades at that point in the pipe system.
The magnitude of the energy losses primarily depends on the shape of the fitting.
The head or energy loss can be expressed by using the applicable resistance coef-ficient for the valve or fitting. The Darcy-Weisbach Equation then becomes:
hm⫽ K (3.10)
K⫽ f (3.11)
Equation 3.10 can be rearranged to express the fitting head loss as feet of straight pipe having the same head loss as the fitting.
Lf⫽ KD (3.12)
To calculate head losses in piping systems with both pipe friction and minor losses use:
hf⫽冢f ⫹冱K冣 (3.13)
Typical K values for the fitting loss coefficients are in Table 3.4.
Table 3.5 lists the estimated pressure drop for thermoplastic lined fittings and valves.
Water Hammer/Pressure Surge
Flowing liquid has momentum and inertia. When flow is stopped suddenly, the mass inertia of the flowing stream is converted into a shock wave. Consequently, a high static head exists on the pressure side of the pipeline. Quick surge pressures are shock waves known as water hammer. Water hammer, or hydraulic transients, is caused by opening and closing (full or partial) valves, starting and stopping pumps, changing pump or turbine speed, reservoir wave action, and entrapped air. The pressure wave from water hammer races back and forth in the pipe, getting pro-gressively weaker with each “hammer.” Maximum surge pressure results when the time required to change a flow velocity a given amount is equal to or less than:
tⱕ 2Lf (3.13)
Tee Standard, flow through run 0.6
Standard, flow through branch 1.8 Notes: Hydraulic Institute, Pipe Friction Manual, 3rd Ed., Crane Company, Technical Paper 410.
where Lf ⫽ is the length of the pipeline, feet
S⫽ is the speed of the pressure wave, feet per seconds t ⫽ is the time, seconds.
S is determined by the following:
(3.14)
where K⫽ Bulk modulus of the liquid, psi (300,000 psi for water) E ⫽ Modulus of elasticity of the pipe material, psi
w⫽ Unit weight of fluid, lb/ft3.
The excess pressure caused by the water hammer can be calculated by:
Ps⫽ (3.15)
where Ps⫽ Change in pressure, psi
vc⫽ Change in velocity, ft/sec, occurring within critical time.
Performing a water hammer analysis of a piping system is a complex task.
Factors to be considered include pumping characteristics, fluid velocity, elevation changes, valve closing times, and piping geometry. Equation 3.15 calculates the maximum surge pressure for the given velocity change. Keeping the time to stop the flow at more than t (Equation 3.13) can minimize pressure changes. The
wSvc
ᎏ144g
FLUID FLOW 3.11
TABLE 3.5 Estimated Pressure Loss for Thermoplastic Lined Fittings and Valves
Size Standard Tee Tee Vertical Horizontal
Inch 90° elbow through through Plug Diaphragm check check
run branch valve valve valve valve
1 1.8 1.2 4.5 2.0 7.0 6.0 16
Notes: Data is for water expressed as equal length of straight pipe in feet.
NA⫽ Part is not available from source.
Source:“Plastic Lined Piping Products Engineering Manual,” page 48.
S⫽
冢 冣冢ᎏwg 144E⫹ᎏKDt 冣
冪莦莦
(144E)Kgreatest effects on the velocity of the liquid occur during the final stage of valve closure. A general guideline for gate valves with linear closure characteristics is to maintain a valve closure time of 10 times t. This should keep the pressure surge at about 10 percent to 20 percent of the surge developed by the t closure time.
Plastic piping materials have different characteristics and handle the effects of pressure surges differently. The designer should consult with the plastic pipe manufacturer concerning their products ability to handle pressure surges. For exam-ple, polyethylene (PE) pipe can handle short-term pressure surges above the design pressure rating of the pipe because of its short-term strength and flexibility. When under similar conditions, surge pressures in PE pipe are significantly less than surges seen in rigid pipe. For the same liquid and velocity change, surge pressures in PE pipe are about 50 percent less than PVC pipe. The fatigue endurance of the plastic piping material must be taken into account if the piping system has fre-quent or continuous pressure surges. A piping system encountering repeated stress could have a long-term strength loss. If the piping system will see frequent cycli-cal surge pressure, the total system pressure (including surge pressure) should not exceed the design pressure rating of the material.