🚀 Orifice Flow Calculator & Simulator
- Set Orifice Diameter (d): Adjust the inner diameter (mm) of the orifice hole, which is reflected in the canvas nozzle aperture.
- Change Head Height (H): Adjust the water level height (m) inside the tank, and observe how a higher head increases the travel distance of the emerging parabolic jet.
- Calibrate Discharge Coefficient (Cd): Enter a precise discharge coefficient suitable for the orifice geometry and machining condition, such as a sharp edge (0.62) or a rounded inlet (0.97).
- Analyze Jet Velocity and Trajectory: Monitor the real-time actual efflux velocity (m/s), discharge LPM, and hourly flow rate on the instrument panel to evaluate the discharge performance.
📚 Orifice Efflux Torricelli’s Law and Fluid Dynamic Definitions of Discharge, Velocity, and Contraction Coefficients ▼
1. Torricelli’s Law and the Theoretical Efflux Velocity Formula
In 1643, Italian mathematician and physicist Evangelista Torricelli discovered that the velocity of a fluid flowing out into the atmosphere through an opening (orifice) in the side or bottom of a tank is geometrically identical to the final velocity of an object free-falling from the height of the fluid surface to the opening (head, H).
v_th = √(2 × g × H)
Where v_th is the theoretical efflux velocity (m/s), g is the acceleration due to gravity (9.80665 m/s²), and H is the tank water level head (m) measured from the center point of the orifice. This equation is a highly idealized mathematical formula derived from Bernoulli’s equation under the assumption that the tank’s cross-sectional area is infinitely larger than that of the orifice (meaning surface velocity is assumed to be zero) and by eliminating the pressure difference.
2. Components of Orifice Coefficients: Discharge (Cd), Velocity (Cv), and Contraction (Cc) Coefficients
In real-world flow, both velocity and flow rate decrease due to the fluid’s viscous friction and inertia as it undergoes abrupt direction changes when passing through the orifice hole. To compensate for this, three fluid coefficient concepts are introduced.
- Coefficient of Velocity (Cv): The reduction ratio of flow velocity caused by internal friction. Defined as
v_act = Cv × v_th, it is typically around0.97 ~ 0.99for standard machined orifices. - Coefficient of Contraction (Cc): After exiting the hole, the flow cross-sectional area becomes slightly smaller than the physical hole area due to inertia, a phenomenon known as the Vena Contracta. It is the ratio of the actual contracted flow area to the hole area, typically around
0.62 ~ 0.64. - Coefficient of Discharge (Cd): A comprehensive coefficient used to calculate the final actual volumetric flow rate (Q), determined as the product of the velocity coefficient and the contraction coefficient.
Cd = Cv × Cc ≈ 0.60 ~ 0.65 (For a sharp-edged orifice)
Q_act = Cd × A_o × √(2gH) [m³/s]
3. Kinematic Trajectory Calculation of the Parabolic Jet
Assuming no air resistance, a horizontally discharging jet exhibits a classic parabolic projectile trajectory, undergoing uniform velocity motion horizontally (x = v × t) and free fall under gravity vertically (y = ½gt²). Given the head height H and the height of the orifice from the ground Y, the maximum horizontal travel distance X is derived by the following optimization formula:
X = 2 × √(H × Y)
This theoretically implies that the water jet reaches its maximum range when the orifice is positioned exactly at the midpoint of the total tank height (H = Y). This fascinating motion is visualized as real-time physics vectors on the simulator canvas.