π Water Hammer (Coup de bΓ©lier) Calculator
Simulate pressure transient waves caused by valve closure using the 1D Method of Characteristics (MOC). Compute Joukowsky wave speeds, check critical closure times, evaluate hoop stress limits, and analyze support forces.
π Configuration
π Simulation Results & Wave Plots
π Results Summary
Wave Speed ($a$)
1,342.9 m/s
Joukowsky relation
Critical Time ($2L/a$)
0.2234 s
Round-trip reflection
Closure Regime
Gradual
Regime classification
Pressure Surge ($\Delta P$)
1,411.9 kPa
14.12 bar / 204.8 psi
Peak Pressure ($P_{max}$)
1,711.9 kPa
17.12 bar
Min Pressure ($P_{min}$)
2.34 kPa
Suction limit
Cavitation Risk
β οΈ CAVITATION RISK
Column separation limit
Force on Support ($F$)
11.09 kN
2,492.8 lbf
Pipe Hoop Stress ($\sigma$)
17.12 MPa
Safety limit check
π Pressure History at Valve (P in kPa vs Time in s)
π‘οΈ Transient Wave Propagation Demo
Watch the pressure wave bounce between the reservoir (fixed boundary) and the valve.
Time step:
0.00s
π¨οΈ Raw Fortran Output
Wave Speed = 1342.86 Critical Time = 0.2234 Closure Type = Gradual DP Inst = 2685.72 DP Grad = 600.00 Pressure Surge = 1411.92 Max Pressure = 1711.92 Min Pressure = 2.34 Cavitation Risk = Yes Support Force = 11.089 Hoop Stress = 17.12 --- TIMELINE DATA --- 0.00000, 244.09 0.00559, 295.24 0.01117, 300.76 0.01676, 306.45 0.02234, 312.27 0.02793, 318.27 0.03351, 324.41 0.03910, 330.74 0.04468, 337.22 0.05027, 343.91 0.05585, 350.76 0.06144, 357.82 0.06702, 365.07 0.07261, 372.54 0.07819, 380.20 0.08378, 388.11 0.08936, 396.23 0.09495, 404.61 0.10053, 413.21 0.10612, 422.09 0.11170, 431.21 0.11729, 440.63 0.12287, 450.31 0.12846, 460.31 0.13404, 470.59 0.13963, 481.22 0.14521, 492.14 0.15080, 503.44 0.15638, 515.06 0.16197, 527.08 0.16755, 539.45 0.17314, 552.24 0.17872, 565.42 0.18431, 579.05 0.18989, 593.10 0.19548, 607.64 0.20106, 622.62 0.20665, 638.13 0.21223, 654.13 0.21782, 670.69 0.22340, 687.78 0.22899, 667.71 0.23457, 681.06 0.24016, 694.81 0.24574, 708.82 0.25133, 723.24 0.25691, 737.94 0.26250, 753.07 0.26808, 768.49 0.27367, 784.36 0.27925, 800.53 0.28484, 817.16 0.29042, 834.11 0.29601, 851.54 0.30159, 869.30 0.30718, 887.54 0.31276, 906.13 0.31835, 925.21 0.32393, 944.65 0.32952, 964.59 0.33511, 984.90 0.34069, 1005.72 0.34628, 1026.90 0.35186, 1048.61 0.35745, 1070.68 0.36303, 1093.27 0.36862, 1116.22 0.37420, 1139.69 0.37979, 1163.51 0.38537, 1187.84 0.39096, 1212.50 0.39654, 1237.67 0.40213, 1263.13 0.40771, 1289.07 0.41330, 1315.29 0.41888, 1341.94 0.42447, 1368.82 0.43005, 1396.10 0.43564, 1423.55 0.44122, 1451.32 0.44681, 1479.21 0.45239, 1488.67 0.45798, 1515.76 0.46356, 1543.16 0.46915, 1570.78 0.47473, 1598.68 0.48032, 1626.74 0.48590, 1655.01 0.49149, 1683.40 0.49707, 1711.92 0.50266, 1706.10 0.50824, 1661.59 0.51383, 1615.87 0.51941, 1568.95 0.52500, 1520.78 0.53058, 1471.35 0.53617, 1420.62 0.54175, 1368.56 0.54734, 1315.16 0.55292, 1260.37 0.55851, 1204.19 0.56409, 1146.56 0.56968, 1087.50 0.57526, 1026.94 0.58085, 964.90 0.58643, 901.32 0.59202, 836.21 0.59760, 769.53 0.60319, 701.30 0.60877, 631.45 0.61436, 560.04 0.61994, 486.98 0.62553, 412.35 0.63111, 336.06 0.63670, 258.22 0.64228, 178.72 0.64787, 97.69 0.65346, 15.03 0.65904, 2.34 0.66463, 2.34 0.67021, 2.34 0.67580, 2.34 0.68138, 2.34 0.68697, 2.34 0.69255, 2.34 0.69814, 2.34 0.70372, 2.34 0.70931, 2.34 0.71489, 2.34 0.72048, 2.34 0.72606, 2.34 0.73165, 2.34 0.73723, 2.34 0.74282, 2.34 0.74840, 2.34 0.75399, 2.34 0.75957, 2.34 0.76516, 2.34 0.77074, 2.34 0.77633, 2.34 0.78191, 2.34 0.78750, 2.34 0.79308, 2.34 0.79867, 2.34 0.80425, 2.34 0.80984, 2.34 0.81542, 2.34 0.82101, 2.34 0.82659, 2.34 0.83218, 2.34 0.83776, 42.22 0.84335, 114.99 0.84893, 189.32 0.85452, 265.30 0.86010, 342.85 0.86569, 422.03 0.87127, 502.74 0.87686, 585.07 0.88244, 597.71 0.88803, 597.70 0.89361, 597.70 0.89920, 597.69 0.90478, 597.69 0.91037, 597.67 0.91595, 597.65 0.92154, 597.62 0.92712, 597.59 0.93271, 597.55 0.93829, 597.50 0.94388, 597.44 0.94946, 597.39 0.95505, 597.33 0.96063, 597.29 0.96622, 597.24 0.97180, 597.21 0.97739, 597.17 0.98298, 597.14 0.98856, 597.12 0.99415, 597.10 0.99973, 597.07 1.00532, 597.06 1.01090, 597.05 1.01649, 597.04 1.02207, 597.03 1.02766, 597.03 1.03324, 597.02 1.03883, 597.02 1.04441, 597.02 1.05000, 597.02 1.05558, 597.02 1.06117, 557.23 1.06675, 484.62 1.07234, 410.45 1.07792, 334.62 1.08351, 257.24 1.08909, 178.22 1.09468, 97.68 1.10026, 15.54 1.10585, 2.93 1.11143, 2.94 1.11702, 2.94 1.12260, 2.95 1.12819, 2.95 1.13377, 2.97 1.13936, 2.99 1.14494, 3.02 1.15053, 3.05 1.15611, 3.09 1.16170, 3.14 1.16728, 3.20 1.17287, 3.25 1.17845, 3.31 1.18404, 3.35 1.18962, 3.40 1.19521, 3.43 1.20079, 3.47 1.20638, 3.49 1.21196, 3.52 1.21755, 3.54 1.22313, 3.57 1.22872, 3.58 1.23430, 3.59 1.23989, 3.60 1.24547, 3.61 1.25106, 3.61 1.25664, 3.62 1.26223, 3.62 1.26781, 3.62 1.27340, 3.62 1.27898, 3.62 1.28457, 43.32 1.29015, 115.77 1.29574, 189.78 1.30132, 265.45 1.30691, 342.67 1.31250, 421.52 1.31808, 501.89 1.32367, 583.84 1.32925, 596.43 1.33484, 596.42 1.34042, 596.42 1.34601, 596.41 1.35159, 596.41 1.35718, 596.39 1.36276, 596.38 1.36835, 596.34 1.37393, 596.32 1.37952, 596.27 1.38510, 596.23 1.39069, 596.16 1.39627, 596.11 1.40186, 596.06 1.40744, 596.01 1.41303, 595.97 1.41861, 595.93 1.42420, 595.89 1.42978, 595.87 1.43537, 595.84 1.44095, 595.82 1.44654, 595.80 1.45212, 595.79 1.45771, 595.77 1.46329, 595.77 1.46888, 595.75 1.47446, 595.75 1.48005, 595.74 1.48563, 595.75 1.49122, 595.74 1.49680, 595.75 1.50239, 595.74 1.50797, 556.13 1.51356, 483.85 1.51914, 409.99 1.52473, 334.48 1.53031, 257.42 1.53590, 178.73 1.54148, 98.54 1.54707, 16.76 1.55265, 4.21 1.55824, 4.22 1.56382, 4.21 1.56941, 4.22 1.57499, 4.23 1.58058, 4.25 1.58616, 4.26 1.59175, 4.29 1.59733, 4.32 1.60292, 4.37 1.60850, 4.41 1.61409, 4.47 1.61967, 4.52 1.62526, 4.58 1.63085, 4.62 1.63643, 4.67 1.64202, 4.70 1.64760, 4.74 1.65319, 4.76 1.65877, 4.80 1.66436, 4.81 1.66994, 4.84 1.67553, 4.84 1.68111, 4.87 1.68670, 4.87 1.69228, 4.88 1.69787, 4.88 1.70345, 4.89 1.70904, 4.88 1.71462, 4.89 1.72021, 4.89 1.72579, 4.89 1.73138, 44.41 1.73696, 116.54 1.74255, 190.24 1.74813, 265.59 1.75372, 342.49 1.75930, 421.01 1.76489, 501.04 1.77047, 582.63 1.77606, 595.16 1.78164, 595.15 1.78723, 595.16 1.79281, 595.14 1.79840, 595.14 1.80398, 595.12 1.80957, 595.11 1.81515, 595.07 1.82074, 595.05 1.82632, 595.00 1.83191, 594.96 1.83749, 594.89 1.84308, 594.85 1.84866, 594.79 1.85425, 594.75 1.85983, 594.70 1.86542, 594.67 1.87100, 594.63 1.87659, 594.61 1.88217, 594.57 1.88776, 594.56 1.89334, 594.53 1.89893, 594.53 1.90451, 594.50 1.91010, 594.50 1.91568, 594.49 1.92127, 594.49 1.92685, 594.48 1.93244, 594.49 1.93802, 594.48 1.94361, 594.49 1.94919, 594.48 1.95478, 555.05 1.96037, 483.08 1.96595, 409.54 1.97154, 334.34 1.97712, 257.60 1.98271, 179.24 1.98829, 99.38 1.99388, 17.97 1.99946, 5.47 2.00505, 5.48 2.01063, 5.47 2.01622, 5.49 2.02180, 5.49 2.02739, 5.51 2.03297, 5.52 2.03856, 5.56 2.04414, 5.58 2.04973, 5.63 2.05531, 5.67 2.06090, 5.74 2.06648, 5.78 2.07207, 5.84 2.07765, 5.88 2.08324, 5.93 2.08882, 5.96 2.09441, 6.00 2.09999, 6.02 2.10558, 6.06 2.11116, 6.07 2.11675, 6.10 2.12233, 6.10 2.12792, 6.13 2.13350, 6.12 2.13909, 6.14 2.14467, 6.14 2.15026, 6.15 2.15584, 6.14 2.16143, 6.15 2.16701, 6.14 2.17260, 6.15 2.17818, 45.49 2.18377, 117.31 2.18935, 190.69 2.19494, 265.73 2.20052, 342.32 2.20611, 420.51 2.21169, 500.20 2.21728, 581.43 2.22286, 593.91 2.22845, 593.89 2.23403, 593.90 2.23403, 593.90
π Calculation Methodology
Mathematical Model & Theory
Water hammer is a pressure surge wave created when a fluid in motion is forced to stop suddenly (e.g., valve closure). The speed of the elastic shock wave $a$ is derived from the **Joukowsky equation** including pipe wall elasticity parameters:
$$a = \frac{\sqrt{K_f / \rho}}{\sqrt{1.0 + \frac{K_f \cdot D}{E_{pipe} \cdot e}}}$$
The maximum theoretical pressure rise for an instantaneous valve closure ($t_c < 2L/a$) is given by:
$$\Delta P_{inst} = \rho \cdot a \cdot V_0$$
If the valve closure is slow or gradual ($t_c \ge 2L/a$), pressure reflections return from the reservoir before the valve is fully closed, mitigating the surge. Rigid column theory estimates this maximum gradual surge as:
$$\Delta P_{grad} = \frac{\rho \cdot L \cdot V_0}{t_c}$$
Academic References
- Wylie, E. B. & Streeter, V. L.: Fluid Transients in Systems, Prentice Hall.
- Chaudhry, M. H.: Applied Hydraulic Transients, Springer.
- Joukowsky, N.: Uber den hydraulischen Stoss in Wasserleitungsrohren, 1898.
Worked Engineering Example
Problem Statement:
Water ($\rho = 1000\text{ kg/mΒ³}, K_f = 2.2\text{ GPa}$) flows through a $L = 150\text{ m}$, $D = 100\text{ mm}$ steel pipe ($E = 200\text{ GPa}$, thickness $e = 5\text{ mm}$) at initial velocity $V_0 = 2\text{ m/s}$. The valve closes in $t_c = 0.5\text{ s}$. Find the wave speed and maximum pressure surge.
Step-by-step Solution:
1. Calculate wave speed $a$:
$$a = \frac{\sqrt{2.2 \times 10^9 / 1000}}{\sqrt{1.0 + \frac{2.2 \times 10^9 \times 0.100}{200 \times 10^9 \times 0.005}}} = \frac{1483.24}{\sqrt{1.0 + 0.22}} = 1342.86\text{ m/s}$$ 2. Calculate critical closure time $t_{crit}$:
$$t_{crit} = \frac{2 L}{a} = \frac{300}{1342.86} = 0.2234\text{ s}$$ 3. Evaluate closure type:
Since $t_c = 0.5\text{ s} > 0.2234\text{ s}$, closure is **Gradual**.
4. Run Method of Characteristics simulation:
Steady state friction results in pressure drops during closing. The simulation solves MOC and yields a maximum pressure surge of **1411.9 kPa** (at $t = t_c = 0.5\text{ s}$), which is higher than the rigid-column approximation (600 kPa) but smaller than the full Joukowsky surge (2685.7 kPa).
Water ($\rho = 1000\text{ kg/mΒ³}, K_f = 2.2\text{ GPa}$) flows through a $L = 150\text{ m}$, $D = 100\text{ mm}$ steel pipe ($E = 200\text{ GPa}$, thickness $e = 5\text{ mm}$) at initial velocity $V_0 = 2\text{ m/s}$. The valve closes in $t_c = 0.5\text{ s}$. Find the wave speed and maximum pressure surge.
Step-by-step Solution:
1. Calculate wave speed $a$:
$$a = \frac{\sqrt{2.2 \times 10^9 / 1000}}{\sqrt{1.0 + \frac{2.2 \times 10^9 \times 0.100}{200 \times 10^9 \times 0.005}}} = \frac{1483.24}{\sqrt{1.0 + 0.22}} = 1342.86\text{ m/s}$$ 2. Calculate critical closure time $t_{crit}$:
$$t_{crit} = \frac{2 L}{a} = \frac{300}{1342.86} = 0.2234\text{ s}$$ 3. Evaluate closure type:
Since $t_c = 0.5\text{ s} > 0.2234\text{ s}$, closure is **Gradual**.
4. Run Method of Characteristics simulation:
Steady state friction results in pressure drops during closing. The simulation solves MOC and yields a maximum pressure surge of **1411.9 kPa** (at $t = t_c = 0.5\text{ s}$), which is higher than the rigid-column approximation (600 kPa) but smaller than the full Joukowsky surge (2685.7 kPa).