TORRICELLI'S THEOREM AND THE ORIFICE EQUATION
In 1843, Scientist and Mathematician Evangelista Torricelli proved that the flow of liquid through an opening is proportional to the square root of the height of the liquid. Torricelli's Theorem (as it is now referred) can be mathematically stated as follows:
Q = A((2gZ)^0.5); where
A = cross sectional area of the opening
g = value of gravity (constant)
Z = height of the liquid
Torricelli's equation can be derived from Bernoulli's theorem, which is mathematically expressed as:
(P/(specific weight of fluid) + Z + V2/2g)1 = (P/(specific weight of fluid) + Z + V2/2g)2; where
P = pressure at a given point in the system
Z = elevation at a given point in the system
V = velocity at a given point in the system
Two points are selected in the tank shown above: (1) at the water surface and (2) at the orifice opening.
We can apply Bernoulli's equation to determine an equation for the velocity of the fluid coming out of the orifice opening.
(P/(specific weight of fluid) + Z + V2/2g)1 = (P/(specific weight of fluid) + Z + V2/2g)2
0 (pressure at water surface) + Z (height of fluid) + 0 (velocity at the water surface) = 0 (pressure at the orifice - jet flow) + 0 (elevation at the orifice) + V2/2g
In rearranging the equation, we arrive at: V = (2gZ)^0.5); WE HAVE DERIVED TORRICELLI'S EQUATION!!!!
The form of Bernoulli's equation that we used to derive Torricelli's Equation neglected any minor or friction losses; We have modeled an IDEAL FLUID FLOW (a fluid which experiences no friction). Scientists have used empirical data to model the flow of REAL FLUID (a fluid which experiences friction) through an orifice. Depending on the shape and contour of the orifice, different discharge coefficients can be applied to Torricelli's Equation to more accurately model the flow through an orifice. The following figure shows some common orifice shapes and their respective discharge coefficients:
The java script shown below models the flow of fluid through a sharp edged orifice. The equations used in the script are as follows:
V = 0.61(2gZ)^0.5)
Q = 0.61A((2gZ)^0.5)
Enter different numbers into the height and cross sectional area boxes and see how the flow and velocity vary with the height of the fluid and cross sectional area of the orifice!!!