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# Bernoulli,s Equation

## Executive summary:

“In order to check the application and validity of Bernoulli’s theorm the flow of air through a convergent-divergent tube was carried out and examined using Pitot-static tube.The readings obtained from the throat section were used as a reference and measurements were carried out across the tube with a spacing of 25mm for the first 200mm and 12.5mm for rest of the length of tube.In order to check whether the experimental results justify the calculations done theoretically using Bernoulli’s theorem or not the total pressure and static pressure were calculated theorotically and experimentally.Since the formation of boundary layer during the experiment was not taken into account while doing theoretical calculations,the results obtained were different from each other.”

## Introduction

The aim of this experiment is to compare the results obtained practically from the apparatus consisting of convergent-divergent tube and the results calculated theoretically from Bernoulli’s equation.By doing this we are actually checking the use of Pitot-static tube and the validity of Bernoulli’s theorem.

## Background

The Bernoulli’s equation describes the motion and the properties related to the motion of a fluid such as pressure, velocity and elevation and it is applicable in areas where steady, incompressible flow is assumed and where there are no frictional forces or are very negligible. This theorem uses the assumption that the frictional effect between the different layers of a flowing fluid known as viscosity is very small as compared to the effects of gravity, inertia and pressure. This assumption is not practical because all fluids have some viscosity. But there are regions across all fluids where the net results of all the frictional forces add up to zero or it is at a minimum level, creating regions known as “regions of inviscid flow” and so it becomes valid for such inviscid flow regions. It is derived using the principle of the conservation of linear momentum.

**P∕ρ + V ^{2} ∕2 + gz = constant**

In the experiment the streamline flow of air is assumed across the convergent-divergent tube and Bernoulli’s theorem is applied to calculate the velocity of the air both theoretically and practically.We also neglect the changes due the potential energy of the system because there is no height change, so the factor **gz**becomes insignificant. The results thus obtained are compared with each other. Rearranging the Bernoulli’s equation to the form:

**V = **√** (****2(Ps – Pt)/ρ****)**

Where,

**Ps = Static pressure, Pt = Total pressure, p = Reference pressure across throat section. **The above relation was used to calculate the velocity theoretically.

For the theoretical calculations, continuity equation was rearranged as follows (with Q as constant):

**V _{1 }/ V_{2 }= B_{2} / B_{1}**

## Method

**Method**

- The temperature and the pressure were noted in the laboratory as 21°C and 95600 Nm
^{-2 }respectively - With the valve approximately half open, the Pitot-static tube was traversed along the centre-line of the duct in steps of 25mm between 0 and 200mm and in steps of 12.5mm between 200mm and 300mm.
- The total pressure, static pressure and the total pressure in the airbox at each step were recorded with the help of multi-tube manometer.
- The results were tabulated in the table.

## Benrouuli's Examples and Demonstration

## Discussion

- The static pressure was obtained using the static hole in the tube whereas to take the total readings the tube was traversed inwards and outwards through the passage.Because of the existence of viscosity between the layers of a flowing fluid in practical life the experimental results obtained were different from the theoretical results.
- The evidence which indicates that there is a formation of boundary layer is indicated by the difference in the value of velocity as calculated theoretically and practically. For instance the theoretical value of velocity is 0.87 m/s whereas in the experiment it was 0.91 m/s at x = 150mm. The estimated value of the thickness of boundary layer is approximately 3 mm and it is because of the fact that the minimum difference between the theoretical and experimental value after the throat section is 3 mm and as we go further down the divergent section this thickness grows.There are no general equations for boundary layer thickness. Specific equations exist for certain types of boundary layer. For a flat-plate, incompressible, laminar boundary layer, the boundary layer thickness is given by

Where as **Re** is Reynolds number (Re) which equals velocity (V) times density (r) times a characteristic length (l) divided by the viscosity coefficient (mu).

**Re = V * r * l / mu**

- The Mach number at the throat of the duct is 0.135. It is reasonable to assume the flow is incompressible because the temperature is not changing which is the only factor in this experiment which can cause compressibility in the fluid.
- If we change the direction of flow then the convergent area of the passage will be large as compared to the divergent area causing more turbulent flow of air when it leaves the divergent section due to which large differences will be observed between the theoretical values and experimental values.
- The sources of error in the experiment are: the eye level not being exactly parallel to the point from which the reading is being taken, if the flow becomes unsteady and if the temperature changes which causes the fluid to become compressible.
- By noting the difference between the theoretical and experimental values it can be checked that there is a formation of boundary layer.
- The length of the divergent section was designed longer than the convergent section to minimize the turbulence in the flow of air and to make it as smooth and as laminar as possible.

## Summary

In this experiment I learnt the application of Bernoulli’s theorm in practical life and also learnt how to use the Pitot-static apparatus to calculate the velocity of a fluid which in this case was air at different sections of the flow.Because air was used so its density was negligible and potential energy was also negligible.I also learnt from the different experimental and theoretical values of velocity that the formation of boundary layer is inevitable in real life. I also learnt that the divergent area should be larger as compared to convergent area in order to get unperturbed and laminar flow of the fluid.I also came to know after this experiment that changes in atmospheric conditions have a significant effect in the experimental results for instance fluid becomes compressible if the outside temperature changes.

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