Help me with Brenouille's equation

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In summary, Brenouille's equation summarizes the relationship between pressure, kinetic energy, and potential energy in a nonuniform pipe. It states that the sum of these values is constant along a streamline, and can be used to calculate the work done by these forces. This equation is derived by considering the forces on the fluid at different points in the pipe and taking into account changes in kinetic and potential energy.
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SCSA

Hi. Can somebody help me with Brenouille's equation. How is it theoretically derived and what are the effects. I have a project on that. Thank You.
 
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Brenouille's equation says that the sum of the pressure (P), the kinetic energy per unit volume and the potential energy per unit volume, has teh same value at all points along a streamline...

Consider a flow through a nonuniform pipe in a time [del]t...the force on the lower end of the fluid will be P,A, where P, is the pressure at the lower...the work done at teh lower end by the fluid behind it is...

W, = F,[del]x, = P,A,[del]x, = P,V

, - 1 like P1 = P,

; - 2 like P2 = P;

similarly...the work done on the fluid on the upper portion in time [del]t is...

W; = -P;A;[del]x; = -P;V

The assumtion is that the pipe is curved and that the force on the fluid at teh top is opposite to its displacement...therefore the work done is negative...the net work done by these forces is...

W = P,V - P;V

Part of this work goes into changing the kinetic energy of the fluid and part goes into changing its gravitational potential...If m is the mass passing by the pipe in the time interval [del]t, then the change in kinetic energy of the volume of fluid is

[del]KE = 1/2mv;^2 - 1/2mv,^2

the change in potential energy is

[del]PE = mgy; - mgy,

We can apply the work-energy theorm ...W = [del]KE + [del]PE

P,V - P;V = 1/2mv;^2 - 1/2mv,^2 + mgy; - mgy,

density = mass/volume ... [rho] = m/V

P, - P; = 1/2[rho]v;^2 - 1/2[rho]v,^2 + [rho]gy; - [rho]gy,

P, + 1/2[rho]v^2 + pgy, = P; = 1/2[rho]v;^2 + [rho]gy;

P = 1/2[rho]v^2 + pgy = constant
 
  • #3
Brenouille's equation says that the sum of the pressure (P), the kinetic energy per unit volume and the potential energy per unit volume, has teh same value at all points along a streamline...

Thanks stranger...
 

1. What is Brenouille's equation?

Brenouille's equation is a mathematical formula that describes the motion of a compressible fluid in a two-dimensional flow field. It is named after the French mathematician Jean le Rond d'Alembert, who first derived it in the 18th century.

2. How is Brenouille's equation used in science?

Brenouille's equation is used to analyze the behavior of fluids, such as air and water, in various situations. It is commonly used in the fields of aerodynamics and hydrodynamics to study the flow of fluids around objects.

3. What are the variables in Brenouille's equation?

The variables in Brenouille's equation include the fluid density, flow velocity, and pressure. These variables are used to calculate the forces and changes in fluid motion.

4. What are the assumptions made in Brenouille's equation?

Brenouille's equation makes several assumptions, including the fluid being incompressible, the flow being steady and irrotational, and the fluid having a constant viscosity. These assumptions help simplify the equation for easier analysis.

5. What are some real-world applications of Brenouille's equation?

Brenouille's equation has many practical applications, including designing aerodynamic structures, predicting weather patterns, and optimizing the performance of turbines and wings. It is also used in the development of various transportation methods, such as airplanes, cars, and boats.

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