System Curves

Tutorial

The total head at any given flow rate consists of two parts: static head and frictional head. The frictional head loss consists of major losses (piping) and minor head losses (component losses). Each of these is described in further detail below. A system curve shows the total system head required over a range of flow rates.

Pressure and Head Relationship

Head is the expression of the energy content of a liquid in reference to any arbitrary datum. It is expressed in units of energy per unit weight of liquid. The measuring unit for head is feet of liquid.

Pressure and head have a liquid have a physical relationship as outlined in equation 1.A.1. These equations utilize a conversion constant and specific gravity (s), which is defined in equation 1.A.2. It is common to use head because the performance of the pump can be shown independent of the specific gravity or density of the fluid pumped.

$$H=\frac{2.31\cdot p}{s}$$

$$H=\frac{0.102\cdot p}{s}$$

where:

• H = head in feet (ft)
• p = pressure in pounds per square inch (psi)
• s = specific gravity (unitless), see equation 1.A.2

Specific gravity (s) is calculated by equation 1.A.2 and values of specific gravity for water and other liquids can be found in section II.

$$s=\frac{{\rho }_{pumpedfluid}}{{\rho }_{water}}$$

where:

• ρ = density, typically in lbm/ft³

Static Head

Static head consists of both the elevation and pressure difference between the supply and destination of the system. This typically does not depend on velocity and is therefore constant for the system curve. This can be calculated using equation 1.A.3.

$$\Delta h_{stat}=(z_{destination}-z_{supply})+\frac{(p_{destination}-p_{supply})}{\rho \cdot g}$$

where:

• Δh_stat is differential static head, feet
• z is elevation, in feet (ft)
• p is pressure, in psi
• ρ is fluid density, in lbm/ft³
• g is gravitational acceleration, 32.2 ft/s²

This equation is based on the supply and destination being tanks or where the velocity of the fluid can be considered zero. If the pressure measurements are taken with a gage where the velocity is not negligible then the velocity pressure (or dynamic pressure) must be added to the pressure gage reading as indicated in equation 1.A.4.

$$p=p_{gauge}+0.5\cdot \rho \cdot v^2=p_{gauge}+\frac{\rho \cdot Q^2}{2\cdot A^2}$$

where:

• p_gauge is pressure measurement at the gage, in psi
• ρ is fluid density, in lbm/ft³
• v is fluid velocity, in ft/s
• Q is volumetric flow rate, in ft³/s
• A is pipe cross sectional area, in ft²

Since the velocity can be different between the two gage measurements, each of the pressures should be converted separately based on the conditions at the gage before using them in equation 1.A.3. Therefore, the pressure at the supply and destination can be defined as detailed in equations 1.A.5 and 1.A.6.

$$p_{supply}=p_{supply,gauge}+\frac{\rho \cdot Q_{supply}^2}{2\cdot A_{supply}^2}$$

$$p_{destination}=p_{destination,gauge}+\frac{\rho \cdot Q_{destination}^2}{2\cdot A_{destination}^2}$$

Note that if the supply and destination are at the same pressure, as is the case when they are open tanks, then the static head is simply the difference in the liquid elevation.

For the following discussion in this section, it is assumed that the supply and destination are tanks which have negligible velocity.

Frictional Head

The head loss due to friction will vary based on flow rate (velocity) and can be calculated for the system components, such as piping, valves, elbows and bends, and end-use equipment, etc. These losses typically vary proportionally to the square of the velocity.

Major Losses

Frictional head losses in pipes can be calculated using the Darcy-Weisbach equation. The Darcy-Weisbach friction factor, f, can be determined using the Colebrook-White equation (defined in Fluid Flow – General).

These equations will approximate the Moody diagram. The friction factor is based on the Reynolds Number (Re), the pipe diameter (D), and the pipe roughness (ε). The pipe roughness is dependent on the type of pipe being used. Other aspects, such as age, fouling, and coatings will also affect the pipe roughness. An example table of typical values for steel pipe materials an be found here. For other pipe materials, see section IV.

The Hazen-Williams equation is another method to determine pipe losses. These values are only valid for water and do not account for temperature or viscosity. These values are a function of pipe material only and are not dependent on Reynolds Number.

Minor Losses

Minor losses in a piping system occur when fluid passes through a fitting, valve, area change, or enters or exits a tank, etc. Any system component that obstructs or changes the direction or pressure of the flow can be considered a minor loss. These are categorized differently than the pipe frictional loss (or major loss). These minor losses can be the dominant system loss.

The loss created by the component is often characterized by a constant, K, and tabulated for several types of components. Head loss is determined by the equation defined in Fluid Flow – General (K values are also tabulated here).

System Curve

Based on these concepts, the total system head at any given flow rate is the sum of the static head and frictional head losses in the system. It, therefore, can be represented using the equation 1.A.7 (where the velocity at the supply and destination is negligible, e.g. they are tanks):

$$\Delta h_{system}=(z_{destination}-z_{supply})+\frac{(p_{destination}-p_{supply})}{\rho \cdot g}+(\frac{fL}{D}+\Sigma K)\cdot \frac{v^2}{2\cdot g}$$

where:

• Δh_system is system head, feet
• z is elevation, in feet
• p is pressure, in psi
• ρ is fluid density, in lbm/ft³
• g is gravitational acceleration, 32.2 ft/s²
• f is friction factor
• L is pipe length, in feet
• D is pipe inside diameter, in feet
• v is average velocity in pipe, in ft/s

Shape of the System Curve

In some systems the frictional losses will be the majority of overall head loss. These systems will have a steeper system curve.

In other systems the elevation change, or static head, will be the majority of the overall head loss. The system curve in this case will start at a higher value at zero flow and will tend to be flatter.

It is important to accurately characterize the system curve to select the correct pump for various operating conditions as the operating point of your system will be dependent on the intersection between the system curve and the pump curve.

System Curve Application

Real-world applications tend to consider a range or family of system curves. This would bracket the range of liquid levels, operating pressures, valve arrangements, etc.

Educational Demonstration System Curve

This demonstrator shows qualitatively how various parameters affect the system curve. Slide the toggle to change system parameters and see how the system curve varies.

Worked Example (U.S. & Metric Units)

Consider the system in Fig. 1.A.3 and develop a system curve for the flows from 0 to 300 GPM.

Determine the Static Head

Using the static head calculation in Eq. 1.A.3, and since both tanks have the same surface pressure, the static head is only dependent on the difference in surface elevation.

$$\Delta h_{stat}=(z_{destination}-z_{supply})=(289ft-24ft)=265ft$$

$$\Delta h_{stat}=(z_{destination}-z_{supply})=(88.09m-7.315m)=80.77m$$

Determine the Pipe Friction and Properties

To simplify this example, we will consider the friction factor to be constant at 0.02. In general, the friction factor would vary as the flow rate (velocity) varies. Additionally, the flow would be laminar for low velocities. These considerations should be taken into account when calculating the pipe losses.

A 4-inch, schedule 40 steel pipe has an inner diameter of 4.026 inches (0.3355 feet). The overall pipe length in this example is 1255 feet.

Determine the Minor or Component Loss

The losses for the components can be found in tables. In this example we have the following:

• Regular flanged elbow (2), K = 0.31 each
• Swing check valve, K = 2.0
• Wedge-disc gate valve, K = 0.17
• Sudden enlargement, K = 1.0

This gives a total K factor equal to 3.79

Using the combined frictional loss equation in Eq. 1.A.7, we can determine the head loss in (feet)

$$\Delta h_f=(\frac{fL}{D}+\Sigma K)\cdot \left(\frac{v^2}{2\cdot g}\right)=(\frac{0.02\times 1255ft}{0.3355ft}+3.79)\cdot \left(\frac{v^2}{2\times 32.2ft/s^2}\right)=1.22\cdot v^2$$

$$\Delta h_f=(\frac{fL}{D}+\Sigma K)\cdot \left(\frac{v^2}{2\cdot g}\right)=(\frac{0.02\times 382.52m}{0.10226m}+3.79)\cdot \left(\frac{v^2}{2\times 9.81m/s^2}\right)=4.01\cdot v^2$$

Determine the System Curve

The system curve can be calculated by varying the flow rate (velocity) using the above values. Combining the static and frictional losses (pipe friction and minor losses) we have equation 1.A.8 describing system head as a function of static head and friction head, which the susequent calculations utilize to calculate system head as a function of velocity.

$$\Delta h_{system}=\Delta h_{stat}+\Delta h_f$$

$$\Delta h_{system}=265ft+1.22\cdot v^2$$

$$\Delta h_{system}=80.77m+4.01\cdot v^2$$

The following can be used to convert a flow rate (Q) in gpm (gallons per minute) to a velocity in ft/s (with the pipe diameter D in inches).

$$v=0.320833\cdot Q\cdot \left(\frac{4}{\pi \cdot D^2}\right)$$

$$v=0.000278\cdot Q\cdot \left(\frac{4}{\pi \cdot D^2}\right)$$

Substituting this in for velocity and using the 4-inch pipe (ID = 4.026 inches) we get the following as the system curve equation as a function of flow rate in gpm.

$$\Delta h_{system}=265ft+7.75e^{-4}\cdot Q^2$$

$$\Delta h_{system}=80.77m+(4.59e^{-03})\cdot Q^2$$

This, then, gives the following system curve data. This is a system that is dominated by the static head (there is a lift of 265 feet compared to little loss in piping and components).

System Curve Data