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FLUID MECHANICS - THEORY

   

In this section, 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.

     
    Bernoulli's Equation


Flow from a Tank


Flow under a Sluice Gate


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

     
    Velocity at a Point using a Pitot Tube


Piezometer and Pitot Tube
Click to view movie (32k)

 

A combination of piezometer and pitot tube can be used to obtain the velocity at a specific point. Static pressure can be measured using a piezometer. The Pitot tube, as shown in the figure, can be used to measure the stagnation pressure. Stagnation pressure is the pressure when the flow has a stagnation velocity (i.e., V = 0). By perfectly aligning the Pitot tube with the flow, the flow will come to a stop at the tip of the Pitot tube, hence providing the stagnation pressure measurement.

Applying Bernoulli's equation between points 1 and 2 as shown in the figure, and canceling the elevations (equal values) gives,

     


Stagnation Point
Click to view movie (42k)

 

     

Since V2 = 0, the velocity at point 1 is

     

where p1 and p2 are known from the height of the fluid column in the piezometer and Pitot tube, respectively.

   
    Energy and Hydraulic Grade Lines


Energy and Hydraulic Grade Lines

 

The energy grade line (EGL) and the hydraulic grade line (HGL) provide a graphical interpretation of Bernoulli's equation. The EGL represents the total head available with respect to a chosen datum (i.e., a reference line, as shown in the figure).

     

The EGL is a constant for frictionless flow where no work or heat is associated with the process. On the other hand, the HGL is the sum of static pressure and elevation head.

     

Sometimes, this is also referred as the piezometric head and is the height a fluid column would rise in a piezometer. For example, the EGL and HGL for frictionless flow in a duct are shown in the figure.