# Velocity of Buoyant Plug in Closed Loop

• I
• erobz
In summary, the conversation revolves around the problem of natural circulation in a closed loop filled with water and the velocity of a buoyant plug in this system. The discussion touches on topics such as the constancy of the buoyant force, the effect of fluid acceleration on buoyancy, and the energy balance in the system. Approaches using both momentum and energy are proposed to solve the problem, and there is also some confusion regarding the definition of the "plug" in the system. Ultimately, the main question remains unanswered: what is the velocity of the buoyant plug in this setup?
erobz
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A problem I was thinking about when talking about natural circulation in a closed loop in another thread:

What is the velocity of the buoyant plug? The loop is filled with water. There are no viscous forces acting on the plug or in the flow. The flow is assumed incompressible.

1) Is the buoyant force constant? I was thinking conservation of energy, but does the fact that the fluid is accelerating with the plug effect it?

$$\int F_b ~dh = m_p g h + \frac{1}{2}m_p v_p^2 + \frac{1}{2}m_w v_p^2$$

with ## v_w = v_p ##
2) I assume its ok to neglect the small change in the fluid center of mass as the plug ascends?

erobz said:
1) Is the buoyant force constant? I was thinking conservation of energy, but does the fact that the fluid is accelerating with the plug effect it?
The acceleration of the fluid and of the plug should be equal and constant. The net force on the plug should also be constant. Those conditions are all consistent with one another.
erobz said:
2) I assume its ok to neglect the small change in the fluid center of mass as the plug ascends?
One can account for the energy balance either by considering the work done by the buoyant force on the plug or by considering the potential energy of the system. Both account for both plug and fluid. Accordingly, both should yield an identical result.

One way to attack the problem is an approach based on momentum. You have a mass of fluid and a mass of plug both accelerating identically under a constant force. This is a simple SUVAT setup:$$a=f/m$$ $$v=at$$Another way is an approach based on energy. You write down an energy balance for potential and kinetic energy with kinetic energy as a function of velocity and potential energy as a function of position.$$KE + PE = C$$ $$\frac{1}{2}m_1v^2 + gm_2s = C$$Here ##m_1## is the total mass of fluid plus plug and ##m_2## is the net mass of the plug after subtracting out buoyancy. ##s## is vertical displacement, of course. Looking at this equation from a different point of view, it is just a work-energy relationship. It is the very familiar work energy relationship that is associated with a mass being accelerated under a constant force.

erobz and Lnewqban
Isn't this similar to a balance with different weights at both ends, naturally seeking a balance state?
You could replace the piston with a few drops of oil or a big bubble of air, I believe.

russ_watters, erobz and jbriggs444
My concern was for the acceleration of the fluid effecting the buoyant force. For instance a submerged buoyant body looses its buoyancy when the fluid it's inside experiences freefall?

erobz said:
My concern was for the acceleration of the fluid effecting the buoyant force. For instance a submerged buoyant body looses its buoyancy when the fluid it's inside experiences freefall?
Without calculation, I am going to trust my intuition on this one. The force on the plug is reduced by the acceleration. But this has no effect on the calculated result. In particular, you get the correct computed acceleration by dividing the "unaccelerated" buoyant force by the total mass of plug plus fluid even though that is not the actual net buoyant force applied on the accelerating plug.

How could the actual buoyant force on the accelerating plug be anything remotely close to the net force accelerating the fluid plus plug?! The net force (buoyancy minus gravity) on the plug has to be enough to accelerate the plug alone. It would not be anywhere near the right figure to account for the motion of plug plus fluid.

Lnewqban
erobz said:
What is the velocity of the buoyant plug?
I'm confused, what is this "plug?" At first I thought you meant a small control mass of the fluid, but it seems you're considering an actual solid plug? That slides axially along the pipe?

Sorry if I'm being dense. It happens.

gmax137 said:
I'm confused, what is this "plug?" At first I thought you meant a small control mass of the fluid, but it seems you're considering an actual solid plug? That slides axially along the pipe?

Sorry if I'm being dense. It happens.
I'm thinking about a solid plug of some low-density material as a first go. It slides up the tube, circulating the water in the loop.

jbriggs444 said:
Without calculation, I am going to trust my intuition on this one. The force on the plug is reduced by the acceleration. But this has no effect on the calculated result. In particular, you get the correct computed acceleration by dividing the "unaccelerated" buoyant force by the total mass of plug plus fluid even though that is not the actual net buoyant force applied on the accelerating plug.

How could the actual buoyant force on the accelerating plug be anything remotely close to the net force accelerating the fluid plus plug?! The net force (buoyancy minus gravity) on the plug has to be enough to accelerate the plug alone. It would not be anywhere near the right figure to account for the motion of plug plus fluid.
I'm thinking it could be a small correction, if any.... If the fluid is static, it has the hydrostatic pressure gradient across the plugs height. I was just curious how that changes given the fluid (the plug is in) is accelerating upwards. I'm not saying your intuition is wrong, I just wan't to understand if there is potential adjustment.

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erobz said:
I'm thinking it could be a small correction, if any.... If the fluid is static, it has the hydrostatic pressure gradient across the plugs height. I was just curious how that changes given the fluid the plug is in is accelerating upwards. I'm not saying your intuition is wrong, I just wan't to understand if there is potential adjustment.
The pressure field in the accelerating fluid is not the same as one would expect for a simple vertical column of accelerating fluid. It is a [nearly] closed loop. The pressure gradient around the closed loop has to add to zero.

We know how much potential energy it takes to change the position of the fluid+plug system by a certain displacement. That tells us the "force" that is causing the circulation. We know the mass of the fluid+plug system. That's it -- the calculation is essentially done at that point.

The intuitive part is that the "force" calculated for this is identical to the buoyant force on a stationary plug in a bath of the fluid at equilibrium.

Lnewqban and erobz

## What is the velocity of a buoyant plug in a closed loop system?

The velocity of a buoyant plug in a closed loop system depends on several factors including the density difference between the plug and the surrounding fluid, the diameter of the loop, and the fluid dynamics within the loop. Generally, the plug will move at a velocity where the buoyant force is balanced by the drag force exerted by the surrounding fluid.

## How does the density difference affect the velocity of the buoyant plug?

The greater the density difference between the buoyant plug and the surrounding fluid, the greater the buoyant force acting on the plug. This increased buoyant force can result in a higher velocity, assuming other factors such as fluid viscosity and loop geometry remain constant.

## What role does fluid viscosity play in the movement of the buoyant plug?

Fluid viscosity affects the drag force experienced by the buoyant plug. Higher viscosity fluids will exert greater resistance to the movement of the plug, thereby reducing its velocity. Conversely, lower viscosity fluids will offer less resistance, allowing the plug to move more quickly.

## Can the geometry of the closed loop influence the velocity of the buoyant plug?

Yes, the geometry of the closed loop can significantly influence the velocity of the buoyant plug. Factors such as the diameter of the loop, the presence of bends or curves, and the overall length of the loop can all impact the flow dynamics and thus the velocity of the plug. For example, narrower or more convoluted loops can increase resistance and reduce velocity.

## Is it possible to calculate the exact velocity of a buoyant plug in a closed loop?

While it is challenging to calculate the exact velocity due to the complex interplay of factors such as buoyant force, drag force, and fluid dynamics, it is possible to estimate the velocity using fluid mechanics equations and empirical data. Computational fluid dynamics (CFD) simulations can also be employed to provide more accurate predictions.

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