Pump system head requirements change with flow rate. This relationship is shown graphically by the system curve, which represents the total head required by a piping system at different flow rates. The total head can be split into static head (elevation or lift-dependent) and dynamic head (velocity or friction-dependent). A system curve will show how sensitive the operating point (the intersection of the system curve and pump curve) is to flow. If the system curve is wrong, the pump will likely be mis-sized — it will run at a flow and head that wasn’t the design point.
A system curve shows the total differential system head (Δhsystem) or head required to be developed by the pump, referred to as pump total head (H). The system head varies as a function of flow rate as illustrated in Fig. 1.B.1. It is important to accurately characterize the system curve in order to select the correct pump for various operating conditions because the operating point of the system will be dependent on the intersection of the system curve and the pump curve as defined in Eq. 1.B.2 and discussed further in the Operating Point section.
Head is the expression of the energy content of a liquid in reference to any arbitrary datum expressed in units of energy per unit weight of liquid. The measuring unit for head is of liquid. Pressure and head of a liquid in a piping system have a physical relationship as described in the following subsection on the Bernoulli equation Eq. 1.B.1 and is also explained in the Pump System Foundational Concepts section with Eq. 1.A.1a and Eq. 1.A.1b. Head may not be intuitive at first, but it is the most useful way of calculating and expressing the energy contained in pump piping systems independent of the fluid density. Refer to this section for additional information on pump total head, pressure and why head is commonly used for system curves.
Based on the conservation of energy, the Bernoulli equation describes the relationship between three energy terms for a fluid that is both incompressible and has no frictional loss and no added energy, such as from heat transfer, chemical reactions or a pump. Defined in Eq. 1.B.1, the head in Bernoulli systems always remains constant (assuming no energy is added or removed at the boundaries), which is a simplification that is addressed in following sections. The components of the energy expressed in the Bernoulli equation are pressure head, velocity head, and elevation head.
$$ h_{1} = h_{2} = ({32.2 \over {1}} · {144 \over 1} · {p_{1} \over {ρ·g}}) + ({v_{1}^2 \over {2·g}}) + ({Z_1}) = ({32.2 \over {1}} · {144 \over 1} · {p_{2} \over {ρ·g}}) + ({v_{2}^2 \over {2·g}}) + ({Z_2})$$
$$ h_{1} = h_{2} = ({p_{1} \over {ρ·g}}) + ({v_{1}^2 \over {2·g}}) + ({Z_1}) = ({p_{2} \over {ρ·g}}) + ({v_{2}^2 \over {2·g}}) + ({Z_2})$$
Where:
Applying the Bernoulli equation to an example system in Fig. 1.B.2, it illustrates that when evaluating the energy at two points in a piping system when frictional losses are ignored, the energy components are simply exchanged (i.e., pressure for potential, potential for velocity, velocity for pressure, etc.) with the total energy or head remaining constant. Calc. 1.B.1a and Calc. 1.B.1b show the total head relative to the datum is equivalent for points 1 and 2 even though the static pressures, elevation heads, and velocities are different. These calculations illustrate the Bernoulli energy transfer with point 2 having lower pressure head, greater elevation head, and greater velocity head than point 1, but having the same total head.
$$ h_{1} = ({32.2 \over {1}} · {144 \over 1} · {p_{1} \over {ρ·g}}) + ({v_{1}^2 \over {2·g}}) + ({Z_1}) = (11.5 ft) + (0.08 ft) + (1 ft)=12.6 ft $$
$$ h_{1} = ({p_{1} \over {ρ·g}}) + ({v_{1}^2 \over {2·g}}) + ({Z_1}) = (3.5 m) + (0.025 m) + (0.305 m) = 3.8 m $$
$$ h_{2} = ({32.2 \over {1}} · {144 \over 1} · {p_{2} \over {ρ·g}}) + ({v_{2}^2 \over {2·g}}) + ({Z_2}) = (7.95 ft) + (2.66 ft) + (2 ft)=12.6 ft $$
$$ h_{2} = ({p_{2} \over {ρ·g}}) + ({v_{2}^2 \over {2·g}}) + ({Z_2}) = (2.43 m) + (0.81 m) + (0.61 m) = 3.8 m $$
The preceding discussion on the Bernoulli equation provides a foundation for determining the system’s total head as a function of velocity or flow rate (Fig. 1.B.1); however, it needs to be expanded because friction losses due to fluid viscosity will exist and pumps will add energy to the system. Fig. 1.B.3 illustrates the pump total head (H) and the frictional head loss in the pump suction piping (hfs) and the pump discharge piping (hfd). The frictional head losses are the result of pipe friction and flow through tank entrances and exits, pipes, fittings, valves, and other end-use equipment in the system as discussed in the Pipe Frictional Losses and Losses in Valves, Fittings and Bends sections.
To determine the pump total head (H) or total differential system head (Δhsystem) we can consider the change in the Bernoulli head from point 1 to point 2 as defined in Eq. 1.B.1 and add the frictional head losses from point 1 to the pump suction flange and from the pump discharge flange to point 2.
Considering the addition of frictional head losses, we get Eq. 1.B.2 describing the pump total head (H), followed by substituting terms and rearranging the grouping of terms. This illustrates that the system head plotted on the system curve and the pump total head (H) will be identical during operation due to energy conservation and therefore, the system operating point will be the flow and head defined by the intersection of the pump and system curve. Refer to Operating Point section for additional details.
$$ H = [Δh_Bernoulli]+[Σh_f] $$ $$ H = [({p_{2} \over {ρ·g}}+{v_{2}^2 \over {2·g}}+{Z_2})-({p_{1} \over {ρ·g}}+{v_{1}^2 \over {2·g}}+{Z_1})]+[h_fs+h_fd]$$ $$ H = [({{p_{2}-p_{1}} \over {ρ·g}})+({{v_{2}^2}-{v_{1}^2} \over {2·g}})+(Z_2-Z_1)]+[h_fs+h_fd]$$
As illustrated in Eq. 1.B.2, the total head (H) is dependent on the differential pressure head (item 1), velocity head (item 2), elevation head (item 3), and summation of frictional head losses (item 4) as summarized below. - Item 1 - Differential pressure head $$ Δh_g=({{p_{2}-p_{1}} \over {ρ·g}}) $$ - Item 2 - Differential velocity head $$ Δh_v=({{v_{2}^{2}-v_{1}^{2}} \over {2·g}}) $$ - Item 3 - Differential elevation head $$ ΔZ=Z_2-Z_1 $$ - Item 4 - Summation of frictional head losses $$ Σh_f=h_{fd}+h_{fs} $$
It is common to group items 1–4 based on being dependent or independent of velocity. Items 1 and 3, pressure head and elevation head, are velocity independent and may be grouped together as differential static head (Δhstat) as illustrated in Eq. 1.B.3. Items 2 and 4 are velocity dependent and may be grouped together as dynamic head (Δhdyn) as illustrated in Eq. 1.B.4. The pump total head (H) can then be defined as shown in Eq. 1.B.5 based on static and dynamic groupings.
$$ Δh_{stat} = ({{p_{2}-p_{1}} \over {ρ·g}})+(Z_2-Z_1) $$
$$ Δh_{dyn} = ({{v_{2}^{2}-v_{1}^{2}} \over {2·g}})+(h_{fd}+h_{fs}) $$
$$ Δh_{system}=H = Δh_{stat}+ Δh_{dyn} $$
The dynamic head varies with flow rate and is grouped separately because it depends on velocity. The frictional head loss portion of the dynamic head can be calculated for the system components, such as piping, valves, elbows and bends, and end-use equipment. These losses typically vary proportional to the square of the velocity.
Major losses due to friction in pipes for incompressible Newtonian fluids can be calculated using the Darcy-Weisbach equation. The Darcy-Weisbach friction factor, f, can be determined using the Colebrook-White equation or using the Pipe Frictional Loss Calculator. The Hazen-Williams equation is another method used to determine pipe losses; however, are typically only valid for water at standard temperature. The Hazen-Williams C-factor is a function of pipe material only and is not dependent on Reynolds number. See the Pipe Frictional Loss section for additional details.
Minor losses due to head losses in fittings, valves, area changes, tank entrances and exits, and end-use equipment are typically calculated based on a resistance coefficient and the velocity head as defined in the Losses in Valves, Fittings and Bends section.
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In some systems frictional losses will be the majority of overall head loss. These systems will have a steeper system curve because the frictional head loss is generally proportional to velocity squared.
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 because the static head is not directly affected by velocity.
Real-world applications tend to consider a range or family of system curves. This brackets the range of liquid levels, operating pressures, valve arrangements, and potential for the pipes fouling over time. The implications of selecting pumps based on a varying system curve are further discussed in the Operating Point section and the educational demonstrator below can be used to get an idea of how changing system parameters will affect the system curve.
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This demonstrator shows qualitatively how various parameters affect the system curve. You can slide the toggle to change system parameters for tank levels, frictional losses, and tank pressure to see how the system curve varies.
Consider the system in Fig. 1.B.6 and develop a system curve for the flows
Determine the Static Head
Using the static head calculation in Eq. 1.B.3, and since both tanks have the same surface pressure, the static head is only dependent on the difference in surface elevation.
$$ Δh_{stat} = ({{p_{2}-p_{1}} \over {ρ·g}})+(Z_2-Z_1)=(0-0)+(289 \,ft - 24 \,ft)=265\, ft $$
$$ Δh_{stat} = ({{p_{2}-p_{1}} \over {ρ·g}})+(Z_2-Z_1)=(0-0)+(88.09 \,m -7.315\,m)= 80.77\,{m} $$
Determine the Dynamic Head Including Frictional Losses
To simplify the pipe frictional head losses (major losses), 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.
For component losses, refer to the Losses in Valves, Fittings and Bends seciton. The minor loss resistance coefficients (K) for Fig. 1.B.6 are as follows:
This gives a total resistance coefficient (ΣK) equal to 3.79
Using the dynamic head equation Eq. 1.B.4, Calc. 1.B.3 combines the major and minor losses in the first term (see the Losses in Valves, Fittings and Bends section Eq. 3.B.2) and results in the dynamic head as a function of velocity.
$$ \Delta h_{dyn} = {({fL \over D} + ΣK) · ({v^2 \over 2·g})} +({v_{2}^2-v_{1}^2 \over 2·g})= {({0.02 × 1255\,ft \over 0.3355\,ft} + 3.79)· ({v^2 \over 2 × 32.2 \,{ft/s^2}})} + (0) = 1.22·v^2$$
$$ \Delta h_{dyn} = {({fL \over D} + ΣK) · ({v^2 \over 2·g})} +({v_{2}^2-v_{1}^2 \over 2·g})= {({0.02 × 382.5\,m \over 0.10226\,m} + 3.79)· ({v^2 \over 2 × 9.81 \,{m/s^2}})} + (0) = 4.01·v^2$$
Determine the System Curve
Per Eq. 1.B.5, Calc. 1.B.4 illustrates the system curve can be calculated as a function of flow rate by adding the differential static head (Calc. 1.B.2) and the dynamic head (Calc. 1.B.3).
$$ Δh_{system}=H = Δh_{stat}+ Δh_{dyn} = 265\,{ft} + 1.22·v^2$$
$$Δh_{system}=H = Δh_{stat}+ Δh_{dyn} = 80.77\,{m} + 4.01·v^2$$
Calc. 1.B.4b can be used to convert a flow rate (Q) in
$$ v = 0.320833·Q·({4 \over \pi ·D^2}) $$
$$ v = 0.000278·Q·({4 \over \pi ·D^2}) $$
Substituting Calc. 1.B.4b in for velocity in Calc. 1.B.4a using the 4-inch pipe we get Calc. 1.B.4c as the system curve equation as a function of flow rate in .
$$ \Delta h_{system} = 265\,{ft} + {{{7.75e^{-4}}}·{Q^2}} $$
$$ \Delta h_{system} = 80.77\,{m} + 4.59e^{-3}·Q^2 $$
Calc. 1.B.4c is used to generate the velocity data in Fig. 1.B.7 and the system head data in Fig. 1.B.8, each as a function of flow rate with the data table following. This system is dominated by the static head. The static head is compared with approximately at the maximum flow rate.
Last updated on August 31, 2026