Fluid mechanics: Rigid column theory

In summary: The author seems to be trying to say that the fanning friction factor is 4, but I'm not sure what they mean by "f".
  • #1
axe34
38
0
Hello

Please see attached page from a textbook. Can someone explain why H = 4f.le.vo^2/2d.g and why delta H is given by the expression in the book? Note that the figure it mentions is on the top of the page. I have tried for days here.

Thanks
 

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  • #2
I am not familiar with rigid column theory and you didn't exactly provide a lot of information to try to decipher it, but this equation looks like it is rather empirical, so it probably isn't derived from first principles. More likely is that it is dimensionally consistent and includes all the factors upon which the answer should depend, so it works provided you know the value of ##\ell_e##.
 
  • #3
You have ##\Delta p = \rho g \Delta H## and Darcy - Weisbach ## {\Delta p\over l} = f_D {\rho\over 2} {v2\over d} ##
Fuirthermore ## f_d = 4 f##, the Fanning friction factor which seems to be the one your book uses.
They combine to $$
\Delta H = 4f {l v^2\over 2dg}
$$for the head loss in the pipe
 
  • #4
I hope you are able to see the book page ok on the attachment.

When the valve closes, it seems to imply that a piezometer at the valve with show height H + ΔH

Thus, if we define positive direction to the right,

∑forces on the fluid in horizontal pipe = m.a

(ρgH + ρ0) - (ρg(H+ΔH) + ρ0) = ρAL. dv/dt where: ρAL is the mass of fluid in the horizontal pipe and ρ0 is atmospheric pressure (A is pipe cross section)

This gives ΔH = - L/g dv/dt which implies that then delta H is positive, then there will be a deceleration of flow. Why has there not been any friction mentioned here?
 
  • #5
Just read the passage again, it says the fluid is frictionless
 
  • #6
axe34 said:
Just read the passage again, it says the fluid is frictionless
In that case I wonder what they mean with ##f##. But then again, that's in a subsequent paragraph ...
 

FAQ: Fluid mechanics: Rigid column theory

What is "Fluid mechanics: Rigid column theory"?

"Fluid mechanics: Rigid column theory" is a mathematical model used to describe the behavior of fluids in a solid, rigid container. It is based on the principles of fluid mechanics and is often used to analyze the movement and pressure of fluids in pipes, channels, and other structures.

What are the assumptions of the rigid column theory?

The rigid column theory assumes that the container holding the fluid is completely rigid, meaning it does not deform or change shape under the pressure of the fluid. It also assumes that the fluid is incompressible, meaning its volume remains constant, and that there are no external forces acting on the fluid.

What is the equation used in rigid column theory?

The main equation used in rigid column theory is the Bernoulli's equation, which describes the relationship between fluid velocity, pressure, and height along a streamline. Other equations that may be used include the continuity equation, which states that the mass flow rate of a fluid remains constant, and the momentum equation, which relates the forces acting on a fluid to its acceleration.

What is the significance of rigid column theory in practical applications?

Rigid column theory is important in many practical applications, such as the design of pipes, pumps, and hydraulic systems. It allows engineers to predict the behavior of fluids in a given system and make informed decisions about the design and operation of these systems. It is also used in the study of weather patterns, ocean currents, and other natural phenomena.

What are the limitations of rigid column theory?

While rigid column theory is useful in many situations, it does have its limitations. It assumes that the fluid is homogeneous and has a constant density and viscosity, which may not always be the case. It also does not account for the effects of turbulence and viscosity, which can significantly affect the behavior of fluids in certain systems. Additionally, it may not be accurate for extreme conditions, such as high pressures or temperatures.

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