Engineering & Scientific
Torricelli Orifice Jet Velocity, Vena-Contracta Flow & Tank Drain Time
Compute sharp-edged orifice efflux velocity, contracted discharge rate Q = C_d * A_o * SQRT(2*g*h), and exact falling-head cylindrical tank drain time between two liquid levels.
Why falling-head tank drain time scales with SQRT(h_1) - SQRT(h_2) rather than volume / Q_initial
Evangelista Torricelli’s Bernoulli efflux law gives ideal jet speed v_ideal = SQRT(2 * g * h) under static liquid head h (meters). Across a real sharp-edged orifice of geometric area A_o, viscous boundary friction reduces speed by C_v (~0.98) while streamline convergence narrows the jet to a vena contracta of area C_c * A_o (~0.62 * A_o), yielding combined discharge coefficient C_d = C_v * C_c (~0.6076) and instantaneous discharge Q(h) = C_d * A_o * SQRT(2 * g * h).
As a constant-area vertical tank A_tank drains from initial head h_1 down to h_2, head—and therefore flow rate—decays continuously! Integrating -A_tank * dh = C_d * A_o * SQRT(2 * g * h) dt yields exact drain time t_drain = (2 * A_tank / (C_d * A_o * SQRT(2 * g))) * (SQRT(h_1) - SQRT(h_2))—exactly TWICE as long to empty completely (h_2 = 0) as dividing initial volume by initial flow Q(h_1)!
| Orifice / Tank Metric | Formula | Example (h1=5 m, Ao=0.01 m2, Atank=2 m2) |
|---|---|---|
| Discharge Coefficient C_d | C_v * C_c | 0.6076 (0.98 * 0.62) |
| Actual Jet Velocity v_jet | C_v * SQRT(2 * 9.81 * h_1) | 9.71 m/s (Ideal = 9.90 m/s) |
| Initial Flow Rate Q(h_1) | C_d * A_o * SQRT(2 * 9.81 * h_1) * 1000 | 60.18 L/s |
| Full Empty Time (5 m -> 0 m) | (2*A_tank / (C_d*A_o*SQRT(19.62))) * SQRT(h_1) | 332.34 s (vs naive 166.17 s) |
Vena-Contracta Volumetric Orifice Flow Rate (L/s)
Multiply C_v * C_c * A_o * SQRT(2 * 9.81 * h_1) by 1,000 to report initial discharge in liters per second.
=ROUND(B4 * B5 * B2 * SQRT(2 * 9.81 * B6) * 1000, 2)Exact Falling-Head Tank Drain Time Formula (Seconds)
Integrate between initial level h_1 (B6) and final level h_2 (B7) with a guard ensuring h_1 >= h_2.
=IF(B6 <= B7, 0, ROUND(((2 * B3) / (B4 * B5 * B2 * SQRT(2 * 9.81))) * (SQRT(B6) - SQRT(B7)), 2))Try the example
Use the original values or change them to test your understanding.