Bernoulli’s Equation

The Bernoulli’s equation is a combination of static pressure and dynamic pressure. This means that it can be used to find fluid head as well as fluid velocity. To view Bernoulli’s equation refer to equation 1.

(Eq 1) $P_1+\frac{1}{2}ρv_1^2+γz_1=P_2+\frac{1}{2}ρv_2^2+γz_2$

P = Pressure

ρ = Density

v = Velocity

γ = Specific Weight

z = Fluid Height

 

While using Bernoulli’s equation the following assumptions are made. First, viscous effects are assumed negligible. Next, the fluid has to have a steady flow, and the fluid should be considered incompressible. Finally, the fluid particles must follow a stream line.

Stream Line

To have an accurate prediction using Bernoulli’s equation the fluid particles must follow a stream line. This means that the fluids follow the same predictable path that particle before it followed, so that the fluid motion can be easily predicted using dynamics. This is also known as laminar flow. If the fluid particles follow random paths instead of a similar path, it is called turbulent flow. Turbulent flow is considered chaos since it is very difficult to predict the fluid motion due to the fact that fluid particles don’t appear to follow a particular stream line.

Parts of the Bernoulli’s Equation

The Bernoulli’s equation is a combination of dynamic and static pressure. The dynamic pressure is represented by equation 2.

(Eq 2)  $P_D=\frac{1}{2}ρv_1^2$

PD = Dynamic Pressure

ρ = density

v = velocity

Combining the initial gage pressure and dynamic pressure will result in the stagnation pressure. Refer to equation 3.

(Eq 3)  $P_{Stag}=P_1+\frac{1}{2}ρv_1^2$

Pstag = Stagnation Pressure

P1 = Initial Gage Pressure

Finally, the static pressure is represented by equation 4. Notice it’s the same equation used to calculate hydrostatic pressure, which means it represents the pressure head.

(Eq 4)  $P_S=γz_1$

PS = Static Pressure

γ = Specific Weight

z1 = Fluid Head

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