# Derivation of the Emptying Time for a buoyant box to drain

• tjholi
In summary, the problem is to calculate the emptying time while considering volume conservation. The relevant equations are Q=A*sqrt(b(H-h(t)) and dh/dt=Q/S. The goal is to obtain h/H=1-(1-t/te)^2, with te=(2S/A*)(H/b)^1/2. The problem involves solving a 1st order ODE, but the individual is struggling with rearranging the equations. Buoyancy is also assumed in the model, with a volume flux equation of Q=A*sqrt(b(H-h(t)).
tjholi
Homework Statement
I am having trouble deriving the expression from the initial equations. (Calculate the emptying time considering Volume conservation)
Relevant Equations
Q=A*sqrt(b(H-h(t)) And we have dh/dt =Q/S (conservation equation) and we have to obtain h/H = 1-(1-t/te)^2 with te= (2S/A*)(H/b)^1/2
Problem Statement: I am having trouble deriving the expression from the initial equations. (Calculate the emptying time considering Volume conservation)
Relevant Equations: Q=A*sqrt(b(H-h(t)) And we have dh/dt =Q/S (conservation equation) and we have to obtain h/H = 1-(1-t/te)^2 with te= (2S/A*)(H/b)^1/2

I know this is solving a 1st order ODE but I am lost at rearranging prior to solving.
This is not to be submitted, I am revising for an exam)
J

Can you please provide the exact word-for-word statement of this problem?

Chestermiller said:
Can you please provide the exact word-for-word statement of this problem?
Hey : "To Calculate the emptying time consider volume conservation"

tjholi said:
Hey : "To Calculate the emptying time consider volume conservation"
Hey, where does buoyancy come in?

Chestermiller said:
Hey, where does buoyancy come in?
Hey Chestermiller, Buoyancy is assumed as it is a model of simplified Displacement ventilation with associated volume flux Q=A*sqrt(b(H-h(t)) with A*being a constant related to the size of the openings.
Thanks

## 1. What is the purpose of deriving the emptying time for a buoyant box to drain?

The purpose of deriving the emptying time for a buoyant box to drain is to determine how long it will take for a box to completely drain of water when it is floating on the surface. This information can be useful for designing and constructing buoyant structures, such as ships or floating platforms, to ensure their stability and efficiency.

## 2. How is the emptying time for a buoyant box calculated?

The emptying time for a buoyant box can be calculated using the equation t = V/A, where t is the emptying time, V is the volume of the box, and A is the cross-sectional area of the opening through which the water drains. This equation is based on the principles of fluid dynamics and takes into account the buoyancy and pressure of the water inside the box.

## 3. What factors can affect the emptying time for a buoyant box?

Several factors can affect the emptying time for a buoyant box, including the shape and size of the box, the density of the water, the viscosity of the liquid, and the size and location of the draining opening. These factors can impact the flow rate of water out of the box, which in turn affects the overall emptying time.

## 4. How can the emptying time for a buoyant box be optimized?

To optimize the emptying time for a buoyant box, one can adjust the size and location of the draining opening, as well as the shape and size of the box itself. Increasing the size of the opening or using a wider, more streamlined box can decrease the emptying time. Additionally, reducing the density of the water or using a less viscous liquid can also improve the emptying time.

## 5. Are there any limitations to the derivation of the emptying time for a buoyant box?

Yes, there are some limitations to the derivation of the emptying time for a buoyant box. The equation used to calculate the emptying time is based on idealized assumptions and may not accurately reflect real-world conditions. Factors such as turbulence, surface tension, and air resistance may also impact the actual emptying time. Additionally, the equation does not take into account the effects of external forces, such as wind or waves, which can also affect the draining process.

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