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Bernoulli's Equation
Bernoulli's equation will be introduced. The details of the derivation are simplified, with attention focused on proper use of the equation. Restrictions on the application of Bernoulli's equation are also clearly stated to avoid misuse of the equation. A velocity measurement device called a Pitot tube will also be presented. In addition, the concept of energy and hydraulic grade lines will be introduced.
fig. 1 Flow from a Tank
fig. 2 Flow under a Sluice Gate
fig. 3 Flow through a Nozzle
In the Conservation of Energy section, it was shown that for a control volume, the energy equation can be simplified to
In many cases, the head loss (mainly due to viscous effects) can be ignored. If there is no pump or turbine in the system, then the equation becomes
This relationship is a form of the Bernoulli's equation. The same relationship, but in a slightly different form, can be derived by applying conservation of momentum to a fluid element along any streamline in the flow, giving
p + ρV2/2 + ρgz = constant along streamline
where p is the static pressure, ρV2/2 is the dynamic pressure, and ρgz is the hydrostatic pressure. Bernoulli's equation provides the relationship between pressure, velocity and elevation along a streamline. It can be applied to solve simple problems, such as flow from a tank (free jets), flow under a sluice gate and flow through a nozzle. Applying Bernoulli's equation between points 1 and 2 as shown in the figures yields,
However, one should realize that Bernoulli's equation is subject to some restrictions, and can only be applied to certain flow situations. The assumptions made in deriving Bernoulli's equation are:
(1) Steady flow
(2) Incompressible flow
(3) Inviscid flow (zero viscosity)
(4) Flow along a streamline
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