Bernoulli’s principle: What do Bernoulli's limitations mean?

We love Bernoulli’s equation. Almost every day we apply this equation in our fluid flow calculations. What is Bernoulli’s principle? What are Bernoulli’s constraints? Can we apply it to a fluid flow if there is an isothermal phase change?

If we go deeper, we will find many unanswered questions.

A short note

Bernoulli's principle states that an increase in the speed of a fluid occurs simultaneously with a decrease in static pressure or a decrease in the fluid's potential energy. Although Bernoulli deduced that pressure decreases when the flow speed increases, it was Leonhard Euler in 1752 who derived Bernoulli's equation in its usual form. This is an offshoot of Newton’s law of conservation of energy. ?The principle is only applicable for isentropic flows: when the effects of irreversible processes (like turbulence) and non-adiabatic processes (e.g., heat radiation) are small and can be

Detail

The Bernoulli equation states that for an ideal fluid (that is, zero viscosity, constant density, and steady flow), the sum of its kinetic, potential, and thermal energy must not change. This constraint gives rise to a predictable relationship between the velocity (speed) of the fluid, its pressure, and its elevation (relative height).

Bernoulli’s equation

General Misconception:

Pressure and velocity change inversely.

Bernoulli’s equation does not say that pressure is inversely proportional. It simply says within its limitations, P + rhoV^2/2 + rho gh = Constant

Limitations of Bernoulli’s equation:

Bernoulli's equation [1] the flow must be steady, i.e., the flow parameters (velocity, density, etc...) at any point cannot change with time [2] the flow must be incompressible – even though pressure varies, the density must remain constant along with a streamline and [3] friction by viscous forces must be negligible.

Pressure has two parts [1] static and [2] dynamic. In incompressible fluids dynamic pressure ‘q’ sometimes called velocity pressure is the quantity defined

q = 1/2 rho V^2

q is dynamic pressure in pascals (i.e., kg?m?1?s?2),

rho is fluid mass density (e.g., in kg/m3, in SI units),

V is flow speed in m/s.

The dynamic pressure can be thought of as the fluid's kinetic energy per unit volume. For incompressible flow, the dynamic pressure of a fluid is the difference between its total pressure and static pressure. From Bernoulli's law, dynamic pressure is given by

p0 -ps = ? rho V^2

p0 is total pressure and ps is static pressure

p0 -ps always remains content and it is independent of total pressure, therefore ? rho V^2, the kinetic energy, always remains constant. An increase or reduction of total pressure is just a switch between dynamic and static pressure.

The kinetic energy can increase only if there is an increase in temperature. For incompressible liquids, at constant density, the kinetic energy can’t change. Hence velocity remains always constant in a steady-state flow for incompressible liquids. This is what Bernoulli’s equation of energy conservation means within all limitations.?

?Pressure = Static pressure + dynamic pressure

Bernoulli’s principle says, p0 -ps = ? rho V^2 .

p0 is total pressure and ps is static pressure. p0 -ps is the dynamic pressure

This means Bernoulli’s principle says that an increase in the speed of a fluid occurs simultaneously with a decrease in static pressure or a decrease in the fluid's potential energy. This further means if there is an increase in kinetic energy there will be an increase in dynamic pressure. This is the fundamental point of Bernoulli’s principle,

?What do Bernoulli’s limitations mean?

Bernoulli's equation assumes

(1) no change of velocity - meaning no pressure drop

(2) no change in density - meaning no work

(3) negligible friction - meaning no significant heat generation and no entropy generation

Therefore,

PdV = 0

TdS = 0 [ S = Entropy]

This means, dH = dU [H = Enthalpy and U = Internal energy]

dG = dU [ G = Gibbs free energy]

dU is pressure change /volume

That makes dG only dependent on pressure

Gibbs free energy = [Work - Pressure] [ Work is negligible]

Taking into consideration of Bernoulli's limitations

dG = - Pressure

Then what is Bernoulli's equation?

Bernoulli's equation assumes the flow is internally reversible. That is only possible if the flow is adiabatic. Reversible ÷ Adiabatic makes the flow Isentropic