Verification Of Bernoulli's Principle.docx

Verification of Bernoulli’s theorem Group-6 Name: Animesh Phatowali

Roll no: 17AE30005

Aim: To verify Bernoulli’s theorem and plot the graph between 𝑝⁄𝑝0 𝑣𝑠 𝑥 ⁄𝐿 𝑎𝑛𝑑 𝑣 ⁄𝑣0 𝑣𝑠 𝑥 ⁄𝐿 Appartus Required:   

Air flow bench Manometer Pitot static probe

Theory: For inviscid, incompressible, irrotational flow Bernoulli’s theorem says that for any two points in the fluid flow, the sum of static pressure, dynamic pressure and potential energy per unit volume remain constant. i.e. 1 𝑝 + 𝜌𝑣 2 + 𝜌𝑔ℎ = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 2 When change in altitude is negligible, the physical significance of Bernoulli’s equation is obvious from equation, when the velocity increases, the pressure decreases and vice versa. 𝑖𝑓 𝑝𝑠 = 𝑠𝑡𝑎𝑡𝑖𝑐 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 1 2 𝜌𝑣 = 𝑑𝑦𝑛𝑎𝑚𝑖𝑐 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 2 𝑎𝑛𝑑 𝑝0 = 𝑡𝑜𝑡𝑎𝑙 𝑠𝑡𝑎𝑔𝑛𝑎𝑡𝑖𝑜𝑛 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑇ℎ𝑒𝑛,

1 𝑝0 = 𝑝𝑠 + 𝜌𝑣 2 2 1

2(𝑝0 − 𝑝𝑠 ) 2 ⇒𝑣=[ ] 𝜌 Pitot static tube: Measurement of airspeed It consists of a static tube, which measures the stagnation pressure. Two or more holes are present on the outer wall of the static probe. The plane of the hole is parallel to the flow, as shown in the figure at point A. Because the flow moves over the opening the pressure felt at point A is due to the random motion of molecules. i.e. at point A the static pressure is measured. Such a hole in the surface is called a static pressure orifice.

Procedure: 1. The free stream, total and static pressure were measured. 2. The pitot probe was moved to different positions in the test section to measure the static and total pressure using manometer. 3. Using Bernoulli’s theorem we can calculate velocity at different positions. 4. Then we verify velocity using continuity equation.

OBSERVATION Gp-1,2 P∞=12.2 , Pr =14.6 ,

Position

Area

Ps (static)

P0(Stagnation)

Velocity

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

37.5 35 33 31 2 26 24 21 21 21 21 21 21 22 23 24 25 26 27 27.5 28 29 30 31 32 32.5 33 34 35 36 37

11.8 11.2 10.8 10.1 9.4 8.6 7.4 6.8 6.4 6.2 6.1 6.1 6.3 6.6 7.0 7.4 7.8 8.1 8.4 8.6 8.9 9.1 9.4 9.5 9.7 9.8 10 10.9 11.4 11.6 11.9

14.4 14.4 14.4 14.4 14.4 14.4 14.4 14.4 14.4 14.4 14.4 14.4 14.4 14.4 14.4 14.4 14.4 14.4 14.4 14.4 14.4 14.4 14.4 14.4 14.4 14.4 14.4 14.4 14.4 14.4 14.4

20.95684 23.24953 24.65985 26.95091 29.06191 31.30063 34.38652 35.82993 36.76073 37.2174 37.44365 37.44365 36.98977 36.29831 35.35534 34.38652 33.38959 32.62191 31.83573 31.30063 30.48039 29.92107 29.06191 28.76982 28.17657 27.8752 27.26249 24.31494 22.51126 21.74794 20.54987

Gp-3,4 P8 = 13 mb , Pr = 16.8,

Position

Area

Ps (in millibar)

Po (in millibar)

Velocity

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

37.5 35 33 31 2 26 24 21 21 21 21 21 21 22 23 24 25 26 27 27.5 28 29 30 31 32 32.5 33 34 35 36 37

12.4 11.6 10.8 10 8.8 7.8 6 4.8 4.2 3.8 3.8 3.8 4.2 4.6 5 5.8 6.4 6.8 7.2 7.6 8 8.4 8.6 9 9.2 9.4 9.6 9.8 10 10.2 10.3

16.6 16.6 16.6 16.6 16.6 16.6 16.6 16.6 16.6 16.6 16.6 16.6 16.6 16.6 16.6 16.6 16.6 16.6 16.6 16.6 16.6 16.6 16.6 16.6 16.6 16.6 16.6 16.6 16.6 16.6 16.6

26.63568 29.06191 31.30063 33.38959 36.29831 38.55498 42.31478 44.64575 45.76674 46.49906 46.49906 46.49906 45.76674 45.02252 44.26578 42.71211 41.50871 40.68667 39.84768 38.99064 38.11434 37.2174 36.76073 35.82993 35.35534 34.87429 34.38652 33.89172 33.38959 32.8798 32.62191

Gp- 5,6 P8 =14 , Pr =18.8 Position

Area

Ps (in millibar)

Po (in millibar)

Velocity

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

37.5 35 33 31 2 26 24 21 21 21 21 21 21 22 23 24 25 26 27 27.5 28 29 30 31 32 32.5 33 34 35 36 37

12.8 12 10.8 9 7.4 6.4 4.6 2.8 1.8 1.4 1.4 1.4 1.8 2.4 3.2 4 4.6 5.4 6 6.6 7 7.6 8 8.4 8.8 9 9.2 9.6 9.8 10 10.4

18.6 18.6 18.6 18.6 18.6 18.6 18.6 18.6 18.6 18.6 18.6 18.6 18.6 18.6 18.6 18.6 18.6 18.6 18.6 18.6 18.6 18.6 18.6 18.6 18.6 18.6 18.6 18.6 18.6 18.6 18.6

31.30063 33.38959 36.29831 40.26936 43.49588 45.39615 48.62988 51.66158 53.27136 53.90181 53.90181 53.90181 53.27136 52.31144 51.00344 49.66101 48.62988 47.22001 46.13435 45.02252 44.26578 43.10578 42.31478 41.50871 40.68667 40.26936 39.84768 38.99064 38.55498 38.11434 37.2174

Discussion: 

Difference between static and total pressure Static pressure is the pressure you have if the field isn’t moving or if you are moving with the fluid. Static pressure is felt when the fluid is at rest when the measurement is taken when travelling along with the fluid flow. It is the force exerted on fluid particle from all directions, and is typically measured with gauges and transmitters attached to the side of a pipe or tank wall. Total pressure is the force per unit area that is felt when a flowing fluid is brought to rest and is usually measured with a pitot tube type instrument. The total pressure is the sum of the static pressure and the dynamic pressure. 𝑃𝑡𝑜𝑡𝑎𝑙 = 𝑃𝑠𝑡𝑎𝑡𝑖𝑐 + 𝑃𝑑𝑦𝑛𝑎𝑚𝑖𝑐



What are the possible sources of error? The assumption behind Bernoulli’s theorem have ben stated above and is responsible for the variation in the observed case from the predicted case. A boundary layer is formed near the walls (where the flow is rotational), while we assume a streamline flow. Moreover, the real flow is compressible and viscous. Body forces (gravity in our case) have also being ignored. According to Bernoulli’s theorem, the total pressure (dynamic+static) remains constant and the plot of pressure vs. distance verifies it. The continuity equation 𝐴𝑣 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 Is not valid in real flow situation, which is compressible and three dimensional, hence the plot is different from expected. We can improve upon this by using a larger channel to ignore the boundary layer.



Working of pitot static tube

The basic pitot tube consists of a tube pointing directly into the fluid flow. As this tube contains fluid, a pressure can be measured; the moving fluid is brought to rest (stagnates) as there is no outlet to allow flow to continue. This pressure is the stagnation pressure of the fluid, also known as the total pressure or (particularly in aviation) the pitot pressure. The measured stagnation pressure cannot itself be used to determine the fluid flow velocity (airspeed in aviation). However, Bernoulli's equation states: Stagnation pressure = static pressure + dynamic pressure Which can also be written 𝜌𝑢2

𝑝𝑡 = 𝑝𝑠 + ( 2 ) Solving that for flow velocity gives 2(𝑝𝑖 − 𝑝𝑠 ) 𝑢=√ 𝜌 where

   

𝑢 is the flow velocity; 𝑝𝑖 is the stagnation or total pressure; 𝑝𝑠 is the static pressure; 𝜌 and is the fluid density.