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Old Oct 8th 2018, 02:38 AM   #1
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First law over constricting pipe section

So i am trying to get a sense of the change in internal energy for a fluid that flows through a constricting pipe ($\displaystyle d_{out}<d_{in}$). Bernouilli won't do since the flow is turbulent, hence it contains irreversibilities which will affect the internal energy.

First law:
$\displaystyle U_{in} + \frac{p_{in}}{\rho}+\frac{u_{in}^2}{2} = U_{out} + \frac{p_{out}}{\rho}+\frac{u_{out}^2}{2}$

$\displaystyle \dot{m}$ remains constant, so using conservation of mass combined with the diameters yields $\displaystyle u_{in}$ and $\displaystyle u_{out}$.

Furthermore, I assume that the system is adiabatic and isolated. $\displaystyle \rho$ is assumed remain constant.

I am looking to get a value for $\displaystyle p_{out}$ and $\displaystyle U_{out}$, two unknowns from one equation...

Am I overlooking another equation? Or how am I to solve this.

thanks in advance!
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Old Oct 8th 2018, 10:14 AM   #2
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Please clarify your naming conventions.
I don't want to have to look up the first law
and work out what is what.
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Old Oct 9th 2018, 03:00 AM   #3
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You forgot the gravitational energy and since your fluid is incompressible: $\displaystyle U_{in}\approx U_{out}$.

It is possible to describe the Bernoulli equation by means of the first law of thermodynamics.
Work on:

General thermal systems


Micro-drop fluid mechanics
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