Finding centerline speed of a flow through a nozzle

In summary, the conversation is about a parabolic equation in the x-direction with two boundary conditions at x=0 and x=L. The equation is u=a+b(x-c)^2 and the values for a and b are determined by the entrance and exit speeds. The individual asking for help is having trouble understanding the equation and asks for clarification on specific points. The expert suggests using Eq. (2) and provides a link for further help with using Latex.
  • #1
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13

Homework Statement


finding an equation for centerline speed.png



Homework Equations



u=a+ b(x-c)^2

We have two boundary conditions at x=0, u=u(entrance) and x=L, u=u(exit)

Source: Fluid Mechanics by Çengel/Cimbala








The Attempt at a Solution



I cannot understand the parabolic equation in the x-direction.
 
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  • #2
relevant equation.png


This is the original solution but it is still hard to understand for me. First I cannot understand general parabolic equation as 1 and why is cet set to 0, a=uentrance and b=(uexit-uentrance)/L^2.

Would you like to explain them?

Thank you.
 
  • #3
mech-eng said:
View attachment 99692

This is the original solution but it is still hard to understand for me. First I cannot understand general parabolic equation as 1 and why is cet set to 0, a=uentrance and b=(uexit-uentrance)/L^2.

Would you like to explain them?

Thank you.
Did you set x = 0? What speed did you obtain?

Did you set x = L? What speed did you obtain there?

What about some intermediate location, say x = L/2? What is the speed there?
 
  • #4
for x=0, u=a+b*(0-c)^2, u=a+b*(c^2)

for x=L, u=a+b*(L-c)^2, u=a+b*(L^2 -2Lc + c^2)

for x=L/2, u=a+b*(L/2 - c)^2, u=a+ b*(L^2 /4 - Lc +c^2)
 
  • #5
mech-eng said:
for x=0, u=a+b*(0-c)^2, u=a+b*(c^2)

for x=L, u=a+b*(L-c)^2, u=a+b*(L^2 -2Lc + c^2)

for x=L/2, u=a+b*(L/2 - c)^2, u=a+ b*(L^2 /4 - Lc +c^2)
No, you use Eq. (2) and make the substitutions there. Eq. (1) is just a generic parabola.
 
  • #6
Meanwhile, would you like to guide me how I can use Latext more effecitively? For example, a link.

Thank you.
 
Last edited:
  • #7
mech-eng said:
Meanwhile, would you like to guide me how I can use Latext more effecitively? For example, a link.

Thank you.
You can use the PF Guide here:
https://www.physicsforums.com/help/latexhelp/

Or you can Google "Latex" for more help on the net.
 

1. How is the centerline speed of a flow through a nozzle determined?

The centerline speed of a flow through a nozzle can be determined by using Bernoulli's equation, which relates the velocity of a fluid to the pressure and height of the fluid. This equation can be solved for the velocity of the fluid at the centerline of the nozzle.

2. What factors affect the centerline speed of a flow through a nozzle?

The centerline speed of a flow through a nozzle is affected by several factors, including the size and shape of the nozzle, the pressure and density of the fluid, and any obstructions or restrictions in the flow path. Changes in these factors can impact the velocity of the fluid at the centerline of the nozzle.

3. Can the centerline speed of a flow through a nozzle be increased?

Yes, the centerline speed of a flow through a nozzle can be increased by increasing the pressure of the fluid or decreasing the size of the nozzle. This can be achieved by using pumps or compressors to increase the pressure, or by using a converging-diverging nozzle design to accelerate the flow.

4. What is the significance of the centerline speed of a flow through a nozzle?

The centerline speed of a flow through a nozzle is an important parameter in fluid mechanics and is used to calculate other important values such as mass flow rate and thrust. It is also crucial in applications such as jet engines, where the velocity of the exhaust gases determines the efficiency and performance of the engine.

5. How is the centerline speed of a flow through a nozzle measured?

The centerline speed of a flow through a nozzle can be measured using various techniques, such as Pitot tubes, hot-wire anemometers, and laser Doppler velocimetry. These methods involve placing a probe in the center of the nozzle and measuring the velocity of the fluid at that point. The most appropriate method depends on the specific application and accuracy required.

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