How to use Bernoulli Equation?

Bernoulli Equation

Bernoulli equation is a type of conservation energy (mechanical) principle. In this section, we discuss mechanical energy.

The simple definition of mechanical energy is—we can convert it into mechanical work directly and completely by using an ideal mechanical device, like an ideal turbine.

Kinetic and potential energies are a form of mechanical energy. Thermal energy is not a type of mechanical energy. Because we can’t convert it into work directly and completely as the second law of thermodynamics suggests.

An important thing to note here is that pressure is not a type of energy. But pressure can produce work. For example, a pressure force acting on a fluid through a distance can cause the fluid to flow. Therefore, we can call it flow-energy.

Now the conservation of energy principle says:

“We can transfer energy into or out of the system by heat or work, and it can only result in the change of energy of that system”.

In other words, we can neither create nor destroy energy.

You can read about Heat and Work here: Heat & Work (First Law)

Where can we apply Bernoulli Equation?

Bernoulli equation, as we discussed, is an energy conservation principle. It tells us that the sum of Kinetic, Potential and Flow-energies of fluid is constant. This principle holds as long as the flow is steady, streamlined, incompressible and inviscid.

Where:

  • The steady flow means, there is no change with respect to time at a particular location.
  • Streamlined means, that all the particles passing from a particular point must follow the same path.
  • Incompressible means, no change in the density of the fluid.
  • Inviscid means the effect of the viscosity is negligible.

But we know that there doesn’t exist any fluid which has zero viscosity. Therefore, we can’t apply the Bernoulli equation everywhere in the fluid flow. However, there may be certain regions where net viscous effects are comparatively smaller than other forces.

The approximation turns out to be reasonable in those particular areas of fluid flow.

For example, the frictional/viscous forces are much larger near the solid walls (boundary layers) and downstream of the bodies.

So we must apply the Bernoulli equation only to the regions where pressure and gravitational forces govern the flow.

Derivation of the Bernoulli Equation

We will use Newton’s Law of Motion to derive the Bernoulli equation. We can also derive it from the first law of thermodynamics (when applied to a steady flow system), but it is beyond our discussion here.

Let us first establish our concept of acceleration and velocity in a streamlined flow.

Acceleration of a Fluid Particle

In a two-dimensional flow, the acceleration has two components. One is along the streamline flow represented by “as” while the other is normal to the direction of streamline. The normal acceleration is an=V2/R.

where “R” is the radius of the curvature. The normal acceleration tells us about the direction.

In a streamlined flow, the radius of the curvature is infinity. Therefore, an=V2/R = 0.

Note: Curvature is the amount by which a curve deviates from being a straight line.

We must not forget that for a steady flow acceleration can never be zero. Steady flow means the velocity doesn’t change at a particular point. However, it can be different at different points.

To determine acceleration, we consider velocity as a function of both time and space, i.e. v(t,s). Therefore, it is important to take the total derivative of a function to see its change when it depends on more than one variable.

Velocity in streamline flow

Since in a steady flow, velocity doesn’t change with respect to time. Therefore,

∂v/∂t = 0.

also

ds/dt = v

⇒ dv/dt= (∂v/∂s)v

Now, we have velocity as a function of ‘s’ only. Therefore, acceleration in the s-direction becomes:

Acceleration in streamline flow
Bernoulli Equation
Bernoulli Equation

Let us consider a cylindrical region of fluid flow. The cross-sectional area of this section is ‘dA’. The force causing the flow is ‘pressure force’, i.e. PdA.

An opposing force also acts on this particular section of the fluid particle. We represent this force as (P+dP)dA.

Where ‘dP’ is the change in pressure force, as the fluid section moves through the distance ‘ds’.

Another force acting on the fluid particle is its weight (W=mg).

Newton’s second law of motion tells us,

ΣFs = mas … (A)

The sum of forces acting on the fluid in the s-direction are:

PdA-(P+dP)dA-mgSinθ.

Sinθ is the only effective component of the weight since Cosθ balances out.

Substituting the value of all the forces and acceleration in Eq. (A).

We have,

Bernoulli Equation

“θ” is the angle between the normal of the streamline and the z-axis at that point. Therefore,

Sinθ= dz/ds

We know that density (ρ) = m/V ⇒ m=ρV

where V=dA.ds

Substituting ρV= ρdAds instead of ‘m’ in equation (1)

Bernoulli Equation

Note: d(v2) = 2vdv (From the power rule of derivative)

which implies:

The last two terms in the above equation are exact differentials. If we consider the flow to be incompressible, the first term also comes out to be an exact differential.

After this assumption, we can integrate the above equation to get,

Bernoulli Equation

This is the Bernoulli equation. The Swiss mathematician Daniel Bernoulli (1700-1782) invented it. An associate of Daniel Bernoulli named Leonhard Euler gave its mathematical interpretation in 1755. It is the conservation of the mechanical energy principle.

The Bernoulli equation tells us that,

“The sum of kinetic, potential and flow-energies remains constant for a fluid flow, provided the flow is streamlined and the frictional effects are negligible. Moreover, the flow needs to be incompressible as well”.

Limitations of Bernoulli Equation

The Bernoulli equation is one of the most versatile equations of fluid mechanics. It is quite simple to use and the same hallmark of the Bernoulli equation makes it susceptible to being improperly used.

It is, therefore, an essential task to understand the limitations of the Bernoulli equation. Otherwise, we may come up with the wrong approximations.

Steady Flow: The Bernoulli equation is only applicable when the flow is steady. Therefore, we must not use it during the start-up or shut-down periods or when the flow is in transient states.

Inviscid Flow: There is no flow which has zero frictional or viscous effects. In the long and narrow passages, the frictional effects are dominant.

Moreover, in the diffusers, the flow separates from the walls. In boundary layers, there is a greater amount of friction.

The frictional effects are also dominant near the surfaces which have sharp edges or form rapidly mixing and backflow such as partially closed valves. All the above situations are not suitable for the application of the Bernoulli equation.

So we must apply it only to the shart flows which have large cross-sections due to which fluid velocity is comparatively smaller and in the core region of the conduit/pipe where flow is streamlined.

No Shaft Work: Since we have already discussed that the Bernoulli equation is applicable to the streamlined flow.

Therefore, we shouldn’t use it near the energy adding or consuming devices, such as fans, pumps, turbines, etc. These devices not only disturb the streamlining but also involve energy transfer.

Read about:

Centrifugal Pumps

Positive Displacement Pumps

Incompressible flow: This is one of the assumptions we used during the derivation of the Bernoulli equation. Incompressible flow means the density of the fluid doesn’t vary.

When the Mach number is <0.3, the density of liquids and gases stays constant.

Minimum Heat Transfer: The temperature has a significant effect on the change in density of the fluid. Therefore, we can’t use the Bernoulli equation in situations that involve a great amount of heat transfer.

Read about:

Heat Exchangers

Streamlined Flow: This is another important assumption, we used during the derivation of the Bernoulli equation. Streamlined flow means that every particle passing through a particular section of the flow must follow the same path, the flow is irrotational and the vorticity (curl of the velocity field) is zero.

Share

One Response

Leave a Reply

Your email address will not be published. Required fields are marked *

This site uses Akismet to reduce spam. Learn how your comment data is processed.

Related articles

Have any questions?

Would you like to talk about it? I’ve got plenty of time to listen.