Bernoulli’s equation -1

Here in this post we will be discussing about the Bernoulli principle, the Bernoulli effect. Bernoulli’s equation is directly the result of the conservation of energy principle. Three are there major conservation principles which govern the physics are conservation of mass, conservation of energy and conservation of momentum. Here we derive Bernoulli’s equation, discuss its practical uses , and solve some practice problems involving Bernoulli’s equation.

Bernoulli’s equation origin

As already told the Bernoulli’s equation is originated from the conservation of energy principle. The meaning of the conservation of the energy is that in straight forward manner it means that ” Energy can neither be created and nor be destroyed”. That means energy is conserved. Here it is better to understand other principles of conservation also in addition to conservation of energy. The other two are also defined in similar way as ” Mass can neither be created nor be destroyed ” for conservation of mass and ” Momentum is conserved in a system when net external force is zero”. The conservation mass has given the continuity equation and the energy conservation has given us the Bernoulli’s equation.

Application of Conservation of Energy Principle

Her we have to understand the process of application of conservation of energy principle when fluid is in motion. Let us consider a section of a pipe line 1-1 to 2-2 wherein fluid is flowing with varying dimensions as shown in below image.

Now let us understand the above simple diagram. The practical working conditions on ground may be more complicated than above. But we simplify the things and understand the principle of conservation of energy from above fluid flow. When fluid flows from section 1-1 to section 2-2 , as per the conservation of energy, the energy is neither created or nor destroyed . It can only be converted from one form to another.

In the entrance of the control volume i.e section 1-1, the diameter of the pipe is $D_1$and the diameter of the pipe at section 2-2 is $D_2$. Let us assume the pipe to be of circular shape. But the principle can be applied to any cross section of the pipe provided we use the factor for converting it into equivalent circular pipe if the pipe cross sectional areas are not known.

Let us assume the fluid enters the pipe section 1-1 with velocity $V_1$ and leaves the pipe section 2-2 with velocity $V_2$. This is a simplified phenomenon. These are mean velocities. Here we are considering a part of pipe between sections 1-1 and 2-2. Here we did not put any mechanical device between sections 1-1 and 2-2. Let the pressure at section 1-1 be $p_1$ and at section 2-2 be $p_2$.

Derivation of Bernoulli’s Equation

From the principle of conservation of energy,

The total energy at section 1-1 = Total energy at section 2-2

At section 1-1 and section 2-2 , the total energy = Kinetic Energy + Potential Energy + Energy due to pressure forces respectively by neglecting effects of other forces.

Change in Energy $\mathrm{dm}(\frac{\Delta p}{\mathit{\rho }})+\mathrm{dm}\frac{\Delta V^2}{2}+\mathrm{dmg}\Delta H$ =0

Here dm = is the infinitesimal mass of fluid.

The first part is cahnge energy due to pressure forces , 2nd part is due to kinetic energy, 3rd part is due to potential energy.

By using the subscript 1 for section 1-1 and subscript 2 for section 2-2 we get,

$\mathrm{dm}(\frac{p_1}{{\mathit{\rho }}_1}-\frac{p_2}{{\mathit{\rho }}_2})+\mathrm{dm}\frac{V_1{}^2-V_2{}^2}{2}+\mathrm{dmg}(H_1-H_2)=0$

Hence , $(\frac{p_1}{{\mathit{\rho }}_1}-\frac{p_2}{{\mathit{\rho }}_2})+\frac{V_1{}^2-V_2{}^2}{2}+g(H_1-H_2)=0$

Therefore, $(\frac{p_1}{{\mathit{\rho }}_{1}g}-\frac{p_2}{{\mathit{\rho }}_{2}g})+\frac{V_1{}^2-V_2{}^2}{2g}+(H_1-H_2)=0$

The final Bernoulli’s Equation is written as :

The above equation is valid for both compressible and in-compressible fluids for steady state flow conditions.

With this derivation of Bernoulli’s equation we close this part -1 and discuss Bernoulli Equation practice problems in Part 2.

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Best of luck to everybody.