Fluid velocity and pipe diameter using the continuity equation

• Bolter
In summary, the conversation discusses a problem related to the continuity equation for incompressible flow. There is confusion about the given numbers and the pressure involved, and it is suggested that the problem may be poorly worded. It is also mentioned that the thread is over a year and a half old.
Bolter
Homework Statement
See below
Relevant Equations
continuity equation
Hi,
Can anyone let me know if I had done this Q correctly?

Thanks for any help!

Delta2
Hmm, something about this problem statement seems wrong. If the pipe opening is of diameter ##d_1 = \mathrm{40mm}## and the fluid enters at ##v_1 = \mathrm{0.5ms^{-1}}##, then the mass flow rate should be $$\dot{m} = A_1 \rho v_1 = \frac{\pi}{4} {d_1}^2 \rho v_1 \approx \mathrm{0.64 kg s^{-1}}$$and not ##\mathrm{5 kg s^{-1}}## like they're saying.

You're right to say that if the kinetic energy density halves, then the speed is reduced by a factor of ##\sqrt{2}##, but apart from that the actual question seems inconsistent to me.

etotheipi said:
Hmm, something about this problem statement seems wrong. If the pipe opening is of diameter ##d_1 = \mathrm{40mm}## and the fluid enters at ##v_1 = \mathrm{0.5ms^{-1}}##, then the mass flow rate should be $$\dot{m} = A_1 \rho v_1 = \frac{\pi}{4} {d_1}^2 \rho v_1 \approx \mathrm{0.64 kg s^{-1}}$$and not ##\mathrm{5 kg s^{-1}}## like they're saying.

You're right to say that if the kinetic energy density halves, then the speed is reduced by a factor of ##\sqrt{2}##, but apart from that the actual question seems inconsistent to me.
I agree with your numbers but I am disturbed by this problem and I am looking for other ways to interpret what is given and what is being asked. One must take the flow rate and fluid density at face value. I know it's a stretch but one might interpret "enters a horizontal pipe of diameter 40 mm with a velocity of 0.5 m/s" to mean that the fluid has velocity 0.5 m/s before it enters the 40 mm pipe. One then would have to calculate the fluid velocity inside the pipe (your calculation) and use that to find the exit fluid velocity and diameter. Also, there is no mention of pressure in the statement. If one is to use the Bernoulli equation additional assumptions need to be made, e.g. the pressure in the fluid before it enters the 40 mm dia. pipe is atmospheric or some such thing. I don't think this is a well written problem unless there is more to this that we don't know. Note that this is part (e) of a multipart problem.

Delta2
@kuruman I think this is just a case of a very poorly worded question about the continuity equation for incompressible flow ##\frac{\partial \rho}{\partial t} = -\nabla \cdot \rho\vec{v}##. There will be a pressure change, but I don't think this is important for this question. If we take a surface that coincides with the pipe and its two faces, then $$\frac{dm}{dt} = -\int_{\partial \Omega} \rho \vec{v} \cdot d\vec{S} = 0$$and by extension, since no fluid crosses the pipe walls, then$$\frac{dm}{dt} = -\rho\vec{v}_{\text{in}} \cdot \vec{S}_{\text{inlet}} - \rho \vec{v}_{\text{out}} \cdot \vec{S}_{outlet} = 0$$in the simple case where ##\vec{S}_{\text{inlet}}## and ##\vec{S}_{\text{outlet}}## are parallel, and the flow uniform across a cross section of the fluid, this is more simply described as$$\rho v_{\text{in}} S_{\text{inlet}} = \rho v_{\text{out}} S_{\text{outlet}}$$i.e. we're simply constraining the net flow of mass into the system to be zero.

...but I think the problem-setter has botched the numbers

kuruman said:
I agree with your numbers but I am disturbed by this problem and I am looking for other ways to interpret what is given and what is being asked. One must take the flow rate and fluid density at face value. I know it's a stretch but one might interpret "enters a horizontal pipe of diameter 40 mm with a velocity of 0.5 m/s" to mean that the fluid has velocity 0.5 m/s before it enters the 40 mm pipe. One then would have to calculate the fluid velocity inside the pipe (your calculation) and use that to find the exit fluid velocity and diameter. Also, there is no mention of pressure in the statement. If one is to use the Bernoulli equation additional assumptions need to be made, e.g. the pressure in the fluid before it enters the 40 mm dia. pipe is atmospheric or some such thing. I don't think this is a well written problem unless there is more to this that we don't know. Note that this is part (e) of a multipart problem.
I mistakenly believed that the pressure would be equal to the water's actual pressure, which is 1000 kg/m3, or?

Kelvin Perry said:
I mistakenly believed that the pressure would be equal to the water's actual pressure, which is 1000 kg/m3, or?
You also mistakenly believe that the units of pressure are kg/m3. Anyway, this thread is about a year and a half old. Or what?

"At the exit of the pipe, the fluid kinetic energy has halved."
compared with when? Before entering the pipe or just after?

kuruman said:
You also mistakenly believe that the units of pressure are kg/m3. Anyway, this thread is about a year and a half old. Or what?
I think he is mistaking pressure for density …

etotheipi said:
...but I think the problem-setter has botched the numbers
Yes, this: it looks like the last sentence was added in error.

1. What is the continuity equation?

The continuity equation is a fundamental principle in fluid dynamics that states that the mass of a fluid entering a section of a pipe must be equal to the mass of the fluid leaving that same section. In other words, the rate of fluid flow must be conserved.

2. How is the continuity equation related to fluid velocity and pipe diameter?

The continuity equation can be used to relate the fluid velocity and pipe diameter in a closed system. By rearranging the equation, we can solve for the velocity of the fluid, which is directly proportional to the pipe diameter. This means that as the diameter of the pipe increases, the velocity of the fluid will decrease, and vice versa.

3. What factors can affect fluid velocity in a pipe?

There are several factors that can affect fluid velocity in a pipe, including the pressure difference between the two ends of the pipe, the viscosity of the fluid, and the roughness of the pipe walls. Changes in these factors can cause variations in fluid velocity, which can be calculated using the continuity equation.

4. How can the continuity equation be used in real-world applications?

The continuity equation is used in a variety of real-world applications, including designing and analyzing water distribution systems, predicting fluid flow in pipelines and channels, and understanding the behavior of fluids in industrial processes. It is also an important tool in the study of aerodynamics and weather patterns.

5. Are there any limitations to the continuity equation?

While the continuity equation is a useful tool for understanding fluid dynamics, it does have some limitations. It assumes that the fluid is incompressible, the flow is steady, and there are no external forces acting on the fluid. In reality, these assumptions may not always hold true, and therefore, the results obtained from the continuity equation may not be entirely accurate.

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