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Column emptying experiment
Definition 
This means that as the column empties and the height $h$ decreases, the velocity $v$ also decreases proportionally.
The key parameters are:
• Inner diameter of the vessel: 93 mm
• Inner diameter of the evacuation channel: 3 mm
• Length of the evacuation channel: 18 mm
These parameters are important to understand and analyze the process of column emptying and how the exit velocity varies with height.
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Viscous liquid column emptying
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Equations
If we consider the profile of ERROR:5449,0 for a fluid in a cylindrical channel, where the speed on a cylinder radio ($v$) varies with respect to ERROR:10120,0 according to the following expression:
involving the tube radius ($R$) and the maximum flow rate ($v_{max}$). We can calculate the maximum flow rate ($v_{max}$) using the viscosity ($\eta$), the pressure difference ($\Delta p$), and the tube length ($\Delta L$) as follows:
If we integrate the velocity across the cross-section of the channel, we obtain the volume flow ($J_V$), defined as the integral of $\pi r v(r)$ with respect to ERROR:10120,0 from $0$ to ERROR:5417,0. This integral can be simplified as follows:
$J_V=-\displaystyle\int_0^Rdr \pi r v(r)=-\displaystyle\frac{R^2}{4\eta}\displaystyle\frac{\Delta p}{\Delta L}\displaystyle\int_0^Rdr \pi r \left(1-\displaystyle\frac{r^2}{R^2}\right)$
The integration yields the resulting Hagen-Poiseuille law:
If there is the pressure difference ($\Delta p$) between two points, as determined by the equation:
we can utilize the water column pressure ($p$), which is defined as:
This results in:
$\Delta p=p_2-p_1=p_0+\rho_wh_2g-p_0-\rho_wh_1g=\rho_w(h_2-h_1)g$
As the height difference ($\Delta h$) is:
the pressure difference ($\Delta p$) can be expressed as:
Flow is defined as the volume the volume element ($\Delta V$) divided by time the time elapsed ($\Delta t$), which is expressed in the following equation:
and the volume equals the cross-sectional area the section Tube ($S$) multiplied by the distance traveled the tube element ($\Delta s$):
Since the distance traveled the tube element ($\Delta s$) per unit time the time elapsed ($\Delta t$) corresponds to the velocity, it is represented by:
Thus, the flow is a flux density ($j_s$), which is calculated using:
If the flow through the tube is described by the equation:
and the pressure difference $\Delta p$ is proportional to the height of the column $\Delta h = h:
we can apply the conservation of flow $J_{V1}=J_V$ between the tube and the column $J_{V2}$:
where the flow in column $J_{V2}$ with cross-sectional area $S$ is given by:
Here, the flux density $j_s$ corresponds to the average velocity, which is equal to the rate of change of height over time:
$j_s = \displaystyle\frac{dh}{dt}$
In this way, we obtain the equation for the height of the column as a function of time:
If in the equation
the constants are replaced by
we obtain the first-order linear differential equation
$\displaystyle\frac{dh}{dt}=\displaystyle\frac{1}{\tau_{hp}} h$
whose solution is
Examples
If there is a column height ($h$) of liquid with the liquid density ($\rho_w$) under the effect of gravity, using the gravitational Acceleration ($g$), the variación de la Presión ($\Delta p$) is generated according to:
This the variación de la Presión ($\Delta p$) produces a flow through the outlet tube with the tube length ($\Delta L$), the tube radius ($R$), and the viscosity ($\eta$) of a volume flow 1 ($J_{V1}$) according to the Hagen-Poiseuille law:
Since this equation includes the section in point 2 ($S_2$), the flux density 2 ($j_{s2}$) can be calculated using:
With this, we obtain:
which corresponds to an average velocity.
To model the system, the key parameters are:
• Interior diameter of the container: 93 mm
• Interior diameter of the evacuation channel: 3.2 mm
• Length of the evacuation channel: 18 mm
The initial liquid height is 25 cm.
The volume flow ($J_V$) can be calculated with the Hagen-Poiseuille law that with the parameters the viscosity ($\eta$), the pressure difference ($\Delta p$), the tube radius ($R$) and the tube length ($\Delta L$) is:
The height difference, denoted by the height difference ($\Delta h$), implies that the pressure in both columns is distinct. In particular, the pressure difference ($\Delta p$) is a function of the liquid density ($\rho_w$), the gravitational Acceleration ($g$), and the height difference ($\Delta h$), as follows:
One of the most basic laws in physics is the conservation of mass, which holds true throughout our macroscopic world. Only in the microscopic world does a conversion between mass and energy exist, which we will not consider in this case. In the case of a fluid, this means that the mass entering through a pipe must be equal to the mass exiting it.
If density is constant, the same applies to volume. In such cases, when we treat the flow as an incompressible fluid, it means that a given volume entering one end of the pipe must exit the other end. This can be expressed as the equality between the flow in Position 1 ($J_1$) and the flow in Position 2 ($J_2$), with the equation:
A flux density ($j_s$) can be expressed in terms of the volume flow ($J_V$) using the section or Area ($S$) through the following formula:
The laminar flow of a fluid with viscosity $\eta$ through a tube of radius $R$ is described by Hagen-Poiseuille's law:
The pressure difference is determined by the height of the column $\Delta h$:
which decreases as the liquid flows out. By applying the continuity equation, we can demonstrate that the height decreases over time as follows:
The characteristic time column with Hagen Pouseuille ($\tau_{hp}$) is calculated from the gravitational Acceleration ($g$), the liquid density ($\rho_w$), the tube length ($\Delta L$), the tube radius ($R$), the section in point 2 ($S_2$), and the viscosity ($\eta$) using:
The column height ($h$), as a function of the time ($t$), exhibits an exponential behavior involving the initial height of liquid column ($h_0$) and the characteristic time column with Hagen Pouseuille ($\tau_{hp}$):
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