# Derivation of Continuity Equation in Cylindrical Coordinates

1. Feb 13, 2007

### 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! =)

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Last edited: Feb 13, 2007
2. Feb 13, 2007

### 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.

3. Feb 13, 2007

### 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! =)

4. Feb 14, 2007

### da_willem

No problem, glad to be of help.

5. Oct 25, 2007

### 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

6. Mar 11, 2008

### 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

7. Mar 11, 2008

### 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

8. Mar 21, 2008

### luben

just wanna be sure, should it be like this below?

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

9. Oct 1, 2008

### montuspatel

Can i get the whole of the corrected derivation please .

10. Nov 28, 2009

### maan_alrawi

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