Derivation of Continuity Equation in Cylindrical Coordinates

jhuleea
Help! I am stuck on the following derivation:

Use the conservation of mass to derive the corresponding continuity equation in cylindrical coordinates.

Please take a look at my work in the following attachments. Thanks! =)

Attachments

• part1.jpg
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Last edited:

da_willem
I think it's in the first attachment where you chose to evaluate only [itex] \rho v_r[/tex] between r and r+dr and not the entire term. You forgot here to include the r, which comes down to taking into account the changing radial velocity but not the change in surface area the fluid flows through.

jhuleea
da_willem,

Thank you thank you thank you! After picking over my work for about a week now, I can't believe I oversaw this rudimentary step!

Your help is greatly appreciated! =)

da_willem
No problem, glad to be of help.

beestinky
"the change in surface area the fluid flows through"

that's the one that had me puzzled for ages, too. once you see it though (or at least once it's pointed out) it seems so obvious.

so thanks from me too

supjames
Help

Hey, I'm doing the same problem and was wondering if you could possibly post the corrected solution to this problem asap.

Thanks,

James

beestinky
in the outward mass flow rate expression posted above, the r-plane surface area expands moving outwards, so the 1st term should be (r+dr) dtheta dz (rho vr)...etc.

fix that, then combine flow rates and divide by dV = dr dz dtheta and you're A for away

luben
just want to be sure, should it be like this below?

$$dV\frac{d\rho}{dt}=dV\frac{\partial\rho}{\partial t}+d\theta dz{[r \rho u_{r}]}\right|^{r+}_{r-}+dr dz{[\rho u_{\theta}]\right|^{\theta +}_{\theta -}}+rd\theta dr {[\rho u_{z}]\right|^{z+}_{z-}}$$

montuspatel
Can i get the whole of the corrected derivation please .

maan_alrawi
i went to know how i can prove the continuity equation of mass transfer in cylindrical coordinates