Tools › Fluids and Heat Transfer › Pipe Flow › Tank Drain and Heat-Up Time › Example
Draining a 3 m by 4 m vertical tank through a 50 mm sharp-edged orifice, and heating 10 m³ of water from 15 to 60 °C with a 6 m² steam coil
| Tool | Tank Drain and Heat-Up Time |
| Source | Merriman, M. (1916). Treatise on Hydraulics, 10th ed. John Wiley and Sons, New York. Art. 32 (emptying a vessel, pp. 69–71), Art. 57 (discharge under a dropping head, pp. 135–136) and Art. 58 (emptying a canal lock, pp. 136–138); U.S. Department of the Interior, Bureau of Reclamation (2001). Water Measurement Manual, 3rd ed., revised reprint. U.S. Government Printing Office, Washington, DC. Equation 2-16 (orifice coefficient 0.61); Hausbrand, E. (1903). Evaporating, Condensing and Cooling Apparatus. Translated by Wright, A. C. Scott, Greenwood and Co., London. Chapter XXI (heating liquids by steam coils, pp. 293–294). |
| Level | Undergraduate |
Problem
A vented vertical cylindrical tank 3 m in diameter is full to 4 m and drains through a 50 mm sharp-edged orifice in its bottom. How long does it take to empty, and how does the level fall with time? Separately, 10 m³ of water is to be heated from 15 to 60 °C by a 6 m² coil fed with steam at 200 kPa gauge; find the heat-up time, the peak duty and the steam used.
Inputs to Enter
- Drain (the default analysis): vertical cylinder D = 3 m, start level 4 m, end level 0; orifice 50 mm, C_d = 0.61; vented; outlet at the tank bottom
- Heat-up (choose the fourth analysis): 10 m³ of water, 998 kg/m³, 4.19 kJ/(kg⋅K), 15 to 60 °C; steam at 200 kPa gauge in a coil of 6 m², U = 800 W/(m²⋅K); loss 1 W/(m²⋅K) on 30 m² to 15 °C ambient
Answers
- Volume 28.27 m³, starting flow Q = 0.61 a √(2 g 4 m) = 10.61 L/s, jet velocity 5.403 m/s
- Drain time t = 2 A √H/(C_d a √(2g)) = 5,330 s = 88.8 min, by the closed form and by numerical integration; exactly twice the 2,665 s the same volume would take at the constant starting flow (Merriman, Art. 32)
- Level after 26.0 min: 2.000 m (half the depth in 29 % of the time); mean outflow 5.304 L/s
- Heat-up: saturation temperature 133.7 °C, latent heat 2,163 kJ/kg (IAPWS-IF97); UA = 4,800 W/K, loss 30 W/K, steady temperature 132.9 °C, time constant 144.3 min
- Time to 60 °C: t = τ ln[(T∞ − T1)/(T∞ − T2)] = 4,160 s = 69.3 min; duty 569.6 kW at the start, 353.6 kW at the end
- Heat stored 522.7 kWh, lost 0.84 kWh; steam condensed 871.4 kg, starting rate 948.1 kg/h
Notes
The tool's default inputs are these two cases; press Submit for the drain, choose the heat-up analysis and press Submit for the batch. Both are illustrative, not printed examples. Checked by test_tank_drain_heat_up.php (529 checks): Merriman's canal lock (80 by 20 ft, 8 ft² of orifices, 16 ft: book 333 s with √(2g) = 8.02, tool 332.4 s), his dropping-head example (13.9 ft³ and 7.61 ft after the first second) and hemispherical-bowl formula, every closed form against scipy quadrature on forty seeded tanks, the pipe outlet, gas pad, pump fill and batch equation against scipy solve_ivp, and Hausbrand's steam example (6,000 kcal; his tabulated mean temperature difference 30.94 °C against the exact logarithmic mean 30.83 °C).