oulli's equation may be applied to a siphon to derive the flow rate and maximum height of the siphon.
- Let the surface of the upper reservoir be the reference elevation.
- Let point A be the start point of siphon, immersed within the higher reservoir and at a depth −d below the surface of the upper reservoir.
- Let point B be the intermediate high point on the siphon tube at height +hB above the surface of the upper reservoir.
- Let point C be the drain point of the siphon at height −hC below the surface of the upper reservoir.
- = fluid velocity along the streamline
- = gravitational acceleration downwards
- = elevation in gravity field
- = pressure along the streamline
- = fluid density
Apply Bernoulli's equation to the surface of the upper reservoir. The surface is technically falling as the upper reservoir is being drained. However, for this example we will assume the reservoir to be infinite and the velocity of the surface may be set to zero. Furthermore, the pressure at both the surface and the exit point C is atmospheric pressure. Thus:
- (Equation 1.)
Apply Bernoulli's equation to point A at the start of the siphon tube in the upper reservoir where P = PA, v = vA and y = −d
- (Equation 2.)
Apply Bernoulli's equation to point B at the intermediate high point of the siphon tube where P = PB, v = vB and y = hB
- (Equation 3.)
Apply Bernoulli's equation to point C where the siphon empties. Where v = vC and y = −hC. Furthermore, the pressure at the exit point is atmospheric pressure. Thus:
- (Equation 4.)
As the siphon is a single system, the constant in all four equations is the same. Setting equations 1 and 4 equal to each other gives:
Solving for vC:
- Velocity of siphon:
The velocity of the siphon is thus driven solely by the height difference between the surface of the upper reservoir and the drain point. The height of the intermediate high point, hB, does not affect the velocity of the siphon. However, as the siphon is a single system, vB = vC and the intermediate high point does limit the maximum velocity. The drain point cannot be lowered indefinitely to increase the velocity. Equation 3 will limit the velocity to a positive pressure at the intermediate high point to prevent cavitation. The maximum velocity may be calculated by combining equations 1 and 3:
Setting PB = 0 and solving for vmax:
- Maximum velocity of siphon:
The depth, −d, of the initial entry point of the siphon in the upper reservoir, does not affect the velocity of the siphon. No limit to the depth of the siphon start point is implied by Equation 2 as pressure PA increases with depth d. Both these facts imply the operator of the siphon may bottom skim or top skim the upper reservoir without impacting the siphon's performance.
Note that this equation for the velocity is the same as that of any object falling height hC. Note also that this equation assumes PC is atmospheric pressure. If the end of the siphon is below the surface, the height to the end of the siphon cannot be used; rather the height difference between the reservoirs should be used.
Although siphons can exceed the barometric height of the liquid in special circumstances, e.g. when the liquid is degassed and the tube is clean and smooth, in general the practical maximum height can be found as follows.
Setting equations 1 and 3 equal to each other gives:
Maximum height of the intermediate high point occurs when it is so high that the pressure at the intermediate high point is zero; in typical scenarios this will cause the liquid to form bubbles and if the bubbles enlarge to fill the pipe then the siphon will "break". Setting PB = 0:
Solving for hB:
- General height of siphon:
This means that the height of the intermediate high point is limited by pressure along the streamline being always greater than zero.
- Maximum height of siphon:
This is the maximum height that a siphon will work. Substituting values will give approximately 10 metres for water and, by definition of standard pressure, 0.76 metres (760 mm or 30 in) for mercury. The ratio of heights (about 13.6) equals the ratio 