8.2 Bernoulli equation

Bernoulli equation

Bernoulli’s equation is an equation of motion. It is an extension of Newton’s second law (force = mass x acceleration). Bernoulli’s equation thus applies regardless of whether or not heat is added during the process. Bernoulli’s equation, for steady, one-dimensional and incompressible flow between stations I and II, becomes:

where

p = pressure (Pa)
ρ = density (kg/m³)
v = flow velocity (m/s)
g = acceleration due to gravity 9,806 (m/s²)
h = height above horizontal datum reference (m)
Δ_pf = pressure loss (Pa)

Figure 8.2a Symbols used in Bernoulli’s equation

The term ρ v²/2 is called the dynamic or kinetic pressure and is sometimes combined with the static pressure p to give the total pressure (stagnation pressure) p_o.

Total pressure = static pressure = dynamic pressure

p_o = p + ρ v²/2 = p + p_dyn       (Equ. 8.2b)

In the case of a loss-free (Δp_f = 0) and horizontal (h_I = h_II) flow, the total pressure remains unchanged. If the dynamic pressure (the velocity) increases, the static pressure will decrease correspondingly. The velocity in a pipe increases as the area (the diameter) reduces.

Figure 8.2b Loss-free, horizontal pipe flow.

Total and static pressure are measured in different ways. A pressure tap at right angles to the direction of flow will sense the static pressure. Ahead of a pitot tube the velocity is reduced and the static pressure rises. At the pitot tube opening, the velocity is equal to zero. The pitot tube thus senses the total pressure in the flow.

As a rule, it is easy to measure the total pressure, whereas the static pressure can easily be effected by measurement errors. A Prandtl tube measures both total and static pressure. The differential pressure, the dynamic pressure, ρ v²/2 is obtained directly.

Figure 8.2c Measuring of static, total and dynamic pressure.

Considering the remaining terms in Bernoulli’s equation, ρ g h corresponds to the static pressure due to a column of liquid of height h. The term Δp_f is the drop in pressure (pressure drop) caused by flow losses due to friction.

In pump technology it is practical to express the Bernoulli equation in terms of head, i.e. meters column of liquid. If all terms in Bernoulli’s equation are divided through by ρg we get:

The various terms are then called:

p/ρg = static head (m)
v²/2g = velocity head (m)
h = height above a horizontal datum (m)
h_f = head loss (Pa)

Since all the terms in Equation 8g refer to height they are easy to illustrate graphically.

Figure 8.2d Graphic illustration of Bernoulli’s equation.