Fluids Mass Balance Integral Question | Homework Statement Explained

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The discussion centers on a mass balance integral related to fluid dynamics, specifically focusing on the equation involving the integral of density over volume and surface. The integral in question is \(\frac{d}{d\,t}(\int_{vol}\rho A_b\,dh) + \int{surf}\rho\sqrt{2gh}\,dA = 0\), where the term \(\sqrt{2gh}\) is pulled through the integral. Participants clarify that while 'h' changes with time, it remains constant with respect to the area 'A', justifying the manipulation of the integral. The conversation confirms understanding of this mathematical step, emphasizing the relationship between variables in the context of fluid mass balance. Overall, the discussion effectively unpacks the reasoning behind the integral manipulation in fluid dynamics.
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Homework Statement



I have include the schematic for visual purposes only. My question is about an integral, so a knowledge of Fluids is not necessarily required to answer my question.

Picture1-40.png



Homework Equations



This is the mass balance in a form that is suitable for this problem:

\frac{d}{d\,t}(\int_{vol}\rho\,dV) + \int_{surf}\rho(\mathbf{v}\cdot\mathbf{n})d\,A = 0

Now the integral that I am concerned with is this next line of the above:

\frac{d}{d\,t}(\int_{vol}\rho A_b\,dh) + \int{surf}\rho\sqrt{2gh}\,dA = 0

Now, in the solution, in the 2nd term, they 'pulled' \sqrt{2gh} through the integral.

I know that 'h' varies with 't' but it does not vary with 'A.' This is why they did that right?

I am just trying to see why that was legal, but I think that I get it. Just another case of
me 'thinking out loud' again. :redface:
 
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Saladsamurai said:
Now, in the solution, in the 2nd term, they 'pulled' \sqrt{2gh} through the integral.

I know that 'h' varies with 't' but it does not vary with 'A.' This is why they did that right?

Yes, you got it. :smile:
 
Thank you! :smile:
 
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