System Head Curve
2.6.2. Dynamic Head
The second component used in generating a system-head curve is the dynamic head. The dynamic head is more complicated as it varies with the flow rate in the pumping system. The dynamic head represents the head loss created by moving flow through the pump station, valves, fittings and piping system. This head loss increases approximately proportional to the velocity rose to a certain power.
For the Darcy Wesibach velocity is raised to the 2, for the Hazen-Williams Equation velocity is raised to the 1.85. This relationship explains why the dynamic system curves always appear as a parabola.
In order to determine the dynamic head for a system, the design engineer should have relevant data regarding the piping network including pipe sizes, lengths and materials, fittings, valves, fluid
properties and flow operating range for the system.
In the water and wastewater industries, two methods are accepted for determining the dynamic losses in a piping system. The two accepted methods are the Darcy-Weisbach and Hazen-Williams Methods. For waterworks application, MWH recommend to use the Hazen-Williams Method because it has been proven to work, is calibrated for water conveyance system and uses the friction factor “C”
established by AWWA M11. Use Darcy-Weisbach for piping velocities higher than 15 fps and for viscous fluid other than water such as chemicals, polymers and others. Both methods are described in detail in the following sections.
2.6.2.1. Hydraulic Losses: Darcy-Weisbach
The Darcy-Weisbach equation is the most often used equation when calculating pipe friction losses.
The equation relates head loss through the pipe to known features of the design, such as the friction factor, length of pipe, velocity of the flow, viscosity, the diameter of the pipe and the force of gravity.
The Darcy-Weisbach equation is: g = acceleration of gravity, 32.2 ft/sec2 f = friction factor, dimensionless
Note, the Darcy Weisbach equation is used to calculate losses associated with the piping only.
Losses associated with fittings are discussed in section 2.6.2.3. At this point in the design, the design engineer has developed the preliminary piping layout therefore; most of this information is available to the design engineer, except for the friction factor.
The friction factor is a variable which depends on the fluid characteristics and the roughness of the pipe. With regards to the fluid characteristics associated with the friction factor, the design engineer must take into account the kinematic viscosity and velocity of the fluid by using the Reynolds Number. The Reynolds (Re) number is a non-dimensional value indicating of the smoothness of flow. At low Reynolds numbers (Re<2100), flow is laminar. As the Reynolds number increases (Re>4000), flow becomes turbulent. The friction factor also takes into account the surface roughness of the material. Depending on the material used for the piping, the smoothness of the pipe inner surface has a direct bearing on the amount of head loss created. Smooth pipe, such as plastics or glass have very low head loss, while concrete pipe has a much higher lead loss. One disadvantage
in using the Darcy Weisbach equation is how to determine the correct roughness factor for the type of pipe or lining material. The Design Engineer is encouraged to consult the pipe manufacturer for the recommended roughness factor (for concrete from 0.001 to 0.01) which have been proven and calibrated in the field. Otherwise if a conservative roughness factor is used, the TDH will be overly conservative. This surface roughness is divided by the diameter of the pipe to determine a relative roughness coefficient.
By using the Reynolds number and the surface roughness of the pipe, the design engineer has enough information to determine the friction factor.
For laminar flow, the friction factor can be directly calculated by using formula below:
64
For turbulent flow, the design engineer has the option of either calculating the friction factor using the Colebrook/White equation or through use of the Moody Diagrams. Both methods provide the same value for
the friction factor. However, the Colebrook/White formula is an implicit equation requiring iterative solution. The Swamee-Jain, shown below, provides direct solution for friction factor value, and is accurate to within 2.5% of Colebrook equation or Moody Diagram. This explicit equation can be used when 5000<Re<3x108 and 10-6 <ε/D<0.01.
0.25
log 3.7 5.74
.
2.6.2.2. Hydraulic Losses: Hazen-Williams
Another method for determining head loss in a pipe is by using the Hazen-Williams Equation. This equation is well recognized and is water and wastewater industry accepted method used for determining head loss in pipes in the water and wastewater industries. This equation initially gained its popularity due to its simplicity compared to the Darcy-Weisbach Equation. The head loss could quickly be calculated without computers or use of the Moody Diagrams. The Hazen-Williams equation relates the pipe friction losses to the system flow rate, pipe diameter and a C-factor. The C factor, determined by experimentation, captures the effects of the roughness of the pipe. The Hazen-Williams Equation is:
10,500 . . . . US Customary Units (gpm, in)
10,700 . . . . SI Units (m3/sec, m)
For the two above equations, hf is given in head loss per 1000 ft (US Customary Units) of length or 1000 m (SI Units) of length
As mentioned previously, the equation utilizes a C factor to account for varying pipe materials and surface roughness. C factors are determined experimentally for various pipe materials, surface roughness values and age of pipe. As can be expected, the C factors are accurate for pipe sizes and flow ranges used in the experiments. This limitation is the one drawback of using the Hazen-Williams equation.
MWH typically requires head loss calculations be performed using Hazen-Williams in lieu of Darcy-Weisbach. Due to the overall ease of using Hazen-Williams, it is good engineering practice for the design engineer to use the Hazen Williams and use Darcy-Weisbach method as a check. The justification for this requirement is the concern over the accuracy of the results. As stated above, the C factors are developed through experimentation for certain size pipes with specific velocities. If the design engineer is performing calculations for pipe sizes or flow velocities outside of this range, there
is the possibility for misleading results with the Hazen-Williams Equation. This is why; the Design Engineer is urged to bracket the head loss between the new and the old pipe. The Hazen-Williams equation should not be used for pipe sizes below 8-inches or above 60-inches. The Darcy-Weisbach Equation, on the other hand, is valid for all pipe sizes and flow velocities as long as the roughness factor is carefully determined.
The design engineer shall pay special attention when using C factors for large diameter pipes. The concern regards the accuracy of the C factors. As the pipe diameters increase, the accuracy of the C factor decreases, therefore the C factor must be adjusted. AWWA Manual M11 describes the
correlation for C factors and pipe diameters.
For smooth internal lining in good condition, the average value of C maybe approximated by the formula
C = 140 + 0.17d
Piping with long term lining deterioration, slime buildup, etc, a lower design value is recommended, as follows
C = 130 + 0.16d Where:
D = inside diameter of piping after lining in inches Comments regarding the use of C Values:
C values of 140 to 150 are suitable for smooth (or lined) pipes larger than 300 mm (12 in).
For smaller smooth pipes, C varies from 130 to 140 depending on diameter
C values from 100 to 150 are applicable in the transitional zone (between laminar and turbulent flow), but the scale effect for different diameters is not included in the formula
The formula is unsuitable and not recommended for old, rough or tuberculated pipes with C values below 100,
Force mains for wastewater can become coated with grease and C valves may vary down to 120 or less for severe grease deposition
C factors and ranges shall be approved by the Chief Mechanical Engineer 2.6.2.3. Hydraulic Losses: Fittings
The Darcy Weisbach and Hazen-Williams equations both address hydraulic losses for the pipe only.
As flow passes through fittings or valves, hydraulic losses are created. The losses are proportional to the velocity, or velocity head, of the flow. The velocity head represents the energy in the system required to move the flow at a specific velocity.
2
2 Where:
V = pipe velocity of the fluid g = acceleration of gravity
As can be expected, the losses experienced at the fittings and valves are directly proportional to this flow energy. This proportionality is described through a K factor. Based on experimentation,
numerous types of valves and fittings have been tested to identify the K factor.
When the design engineer is calculating the system hydraulics, the dynamic losses need to include the fitting losses. A summation of K factors for each experiencing the same flow rate and velocity will provide information regarding the system losses. See the MWH Pump Station Evaluation Tool for more information regarding the use of K factors.