Which equations recommend to model a flux through a Bernoulli tube? The velocity of the flow varies very little along the tube, in addition the tube has different transverse areas. I am doing a project about fluid mechanics, using the Bernoulli equation to calculate pressures through different transverse areas, the water rises through small vertical tubes reaching a certain height corresponding to the pressure generated by the passage of water with a certain speed


Based on your description, the only equations you will need are:
1.) Continuity: $$\dot{m}_1 = \dot{m}_2 \quad \text{or} \quad \rho_1 V_1 A_1 = \rho_2 V_2 A_2$$ where $\dot{m}$ is the mass flow rate through a given station, $\rho$ is the density of the fluid at that station (a constant for incompressible flow), $V$ is the velocity of the fluid, and $A$ is the cross-sectional area of the tube.
2.) Bernoulli: $$p_1 + \frac{1}{2} \rho V_1^2 = p_2 + \frac{1}{2} \rho V_2^2$$ where $p$ is the pressure of the fluid in the tube at a given station. The above expression assumes incompressible, steady, and inviscid flow.
3.) Hydrostatic: $$ p = \rho_m g h $$ where $\rho_m$ is the manometer fluid, $g$ is the gravitational acceleration on earth, and $h$ is the height of the manometer fluid column. This will give the pressure at a specific location in a tube relative to a manometer fluid height change.
Here is a conventional setup: enter image description here

In this image, $Q$ represents the volumetric flow rate, and is simply, $Q = A \cdot V$. A combination of all the above equations yields the classical result: $$ V_2 = \sqrt[]{\frac{2\Delta p}{\rho \left[1-\left(\frac{A_2}{A_1}\right)^2\right]}} $$


$$ \dot{m} = A_2 \sqrt[]{\frac{2 \rho \Delta p}{ \left[1-\left(\frac{A_2}{A_1}\right)^2\right]}} $$

Now consistent with the image, provided the fluid in motion is a gas (i.e. negligible change in pressure due to elevation change), the $\Delta p$ is given by,

$$ \Delta p = \rho_m g h $$

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