Parameters
Centrifugal
Design Equations
Pump curve (parabolic):
$H_p = H_0\!\left(1 - \left(\tfrac{Q}{1.2Q_0}\right)^2\right)$
System curve:
$H_s = H_{s0}+ k_s Q^2$
Shaft power: $P = \dfrac{\rho g Q H}{\eta}$
Specific speed: $N_s = N\sqrt{Q}/ H^{3/4}$
Theory & Key Formulas
$$H = \frac{u_2^2 - u_1^2}{2g} + \frac{p_2 - p_1}{\rho g} + \frac{v_2^2 - v_1^2}{2g}$$
ポンプの全揚程 \(H\) [m]：Euler方程式より導出
$$N_s = \frac{n\sqrt{Q}}{H^{3/4}}$$
比速度 \(N_s\)：ポンプ形式の選定指標（遠心: 100〜400、軸流: 400〜1200）
$$P = \frac{\rho g Q H}{\eta_p}$$
軸動力 [W]：\(\eta_p\) ポンプ効率
What is a Pump Operating Point?
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What exactly is the "operating point" for a pump? I see it mentioned in the simulator description.
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Basically, it's the one flow rate and pressure head where your pump and your piping system are in perfect agreement. The pump wants to push fluid at a certain pressure for each flow rate (its H-Q curve), and the pipes resist the flow, requiring more pressure for higher flow (the system curve). The operating point is where these two curves cross. Try moving the "Pump Design Head" slider above—you'll see the blue pump curve shift and the operating point move instantly.
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Wait, really? So the system curve is just about the pipes? What if I need to pump water up to a higher tank?
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Great question! The system curve has two main parts. The first is the static head—that's the fixed height you need to lift the fluid, like to the top of that tank. The second is the dynamic head from pipe friction, which increases with the square of the flow rate. In the simulator, you control the static head with its own slider. Increase it, and you'll see the entire orange system curve jump upward, forcing a new, lower-flow operating point.
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So the pump's power and efficiency depend on this one point? What happens if my actual operating point is far from the pump's best efficiency point?
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Exactly right. Running far from the Best Efficiency Point (BEP) is a common practical problem. It causes excessive vibration, cavitation (bubbles forming and collapsing), and wasted energy. The simulator calculates shaft power in real-time. Try adjusting the "Pipe Friction Factor" to make the system curve steeper. You'll see the operating point shift left or right, and the calculated power and efficiency will change, showing you the real cost of a mismatched system.
Physical Model & Key Equations
The pump's performance is modeled by a parabolic H-Q curve, which is an approximation of real pump test data. The head produced decreases as the flow rate increases.
$$H_{pump}= H_0 - k Q^2$$
Where:
$H_{pump}$ is the total dynamic head provided by the pump (m).
$H_0$ is the shut-off head (head at zero flow, controlled by the "Pump Design Head" slider).
$k$ is the pump curve coefficient (shape factor).
$Q$ is the volumetric flow rate (m³/s).
The system curve represents the total head the piping network requires. It combines a constant static lift with head losses due to friction, which vary with the square of the flow rate.
$$H_{system}= H_{static}+ C_f Q^2$$
Where:
$H_{system}$ is the total head required by the system (m).
$H_{static}$ is the static head (lift height, controlled by its slider).
$C_f$ is the system friction coefficient (controlled by the "Pipe Friction Factor" slider).
The operating point is found by solving $H_{pump}= H_{system}$ for $Q$.
Real-World Applications
Building HVAC Systems: Centrifugal pumps circulate chilled or hot water through miles of piping in skyscrapers. Engineers use this exact analysis to select a pump that hits the BEP at the building's design flow rate, minimizing electricity costs for the life of the building. A mismatch can lead to noisy operation and high utility bills.
Water Treatment Plants: Pumps move raw water into the plant and treated water out to the municipal network. The system curve changes as filters get clogged or demand peaks. Understanding the operating point allows operators to throttle valves or switch pumps to stay in an efficient and safe operating zone.
Industrial Cooling Circuits: In a factory, pumps circulate cooling water to machinery like injection molders or generators. If a pipe gets restricted (increasing $C_f$), the operating point shifts, potentially reducing flow below safe levels and causing equipment to overheat. Real-time monitoring compares actual vs. predicted operating points.
Irrigation and Agriculture: Pumping water from a well or canal through long, branching irrigation lines is a classic application. The static head is the lift from the water source, and the friction is high due to long pipe runs. Farmers must select a pump that provides enough flow at the end of the line without requiring excessive, costly power.
Common Misconceptions and Points to Note
When you start using this tool, there are a few common pitfalls to watch out for. First, mistaking the "rated point" for the "constant operating point". The rated performance listed in catalogs is ultimately an ideal design point. The actual operating point can deviate significantly depending on the piping. For example, even if you select a pump with a rated flow of 100m³/h, it's not uncommon for the actual flow to be around 70m³/h if the pipe is too narrow. When you manipulate the system curve in the tool, this discrepancy becomes immediately clear.
Next, how to interpret shaft power. The calculated shaft power is a theoretical value for pumping water alone. In a real machine, you have mechanical friction losses and motor efficiency, so it's standard practice to estimate the actual required power as 20-30% higher. If the shaft power is 10kW, you would typically select a motor capacity of 12.5kW or 15kW.
Finally, interpreting specific speed. Specific speed is super useful for "categorizing" pump types, but the calculated value itself isn't an absolute indicator of good or bad performance. For instance, a specific speed around 500 often corresponds to the most efficient radial (volute) type. However, when this value exceeds 800, there's a trade-off: cavitation becomes more likely. When you increase the rotational speed in the tool to raise the specific speed, imagine how the pump's "characteristics" change even for the same head and flow rate.
Worked Example
A centrifugal pump with H-Q curve point at Q=150 m³/h, H=28 m, efficiency=82% serving a system requiring 25 m static head plus 3 m friction loss at design flow. The system curve intersects the pump curve at approximately Q=140 m³/h and H=28 m. Shaft power calculation: P = (ρ×g×Q×H)/η = (1000×9.81×140/3600×28)/0.82 ≈ 44.2 kW. Specific speed: Ns = (N×√Q)/H^0.75 for a 1450 rpm motor yields approximately 62, indicating radial-flow classification suitable for medium heads.