Flux Averaged Concentration

dburnell · Post by **dburnell** » Fri Nov 16, 2012 9:52 pm

For an 1-D infinite domain and continuous source at x=0 with first order decay, does he transformation from volume averaged concentration (Cr)to flux averaged concentration (Cf) have a sign change for x<0 so that Cf = D/v dCr/dX - Cr?

Thanks!

ntoride · Post by **ntoride** » Mon Nov 19, 2012 6:48 am

I think the definition of Cf is same for x <0 as for x>0. Toride et al. (SSSAJ, 1406-1409, 1993) might help you to understand Cf.
Nobuo

rvang · Post by **rvang** » Mon Nov 19, 2012 3:07 pm

Dear dburnell,

No, there is no change in the sign of the concentration. The flux-averaged concentration simply corrects the volume-averaged concentration for the diffusion/dispersion contribution Cf=Cr-D/v dCr/dx.

All the best. --Rien van G.

dburnell · Post by **dburnell** » Mon Nov 19, 2012 9:19 pm

Dear Rien and Nobuo,

Thanks! Using the definition, is it correct to say that the flux concentration is negative for x < 0 for an infinite domain analytical solution of a 1-D continous point source at x=0 with advection, dispersion, and first-order decay?

It seems like it is just a sign convenction because the dispersive flux vector is opposite in direction to the advective flux vector for x<0 in this case

Regards,
Dan Burnell

ntoride · Post by **ntoride** » Tue Nov 20, 2012 1:30 am

Dear Dan,

Cf is the ratio of the solute flux to the water flux. For steady water flow with constant theta (v is constant everywhere), for example, the Cf distribution (Cf vs. position) is identical to the solute flux distribution. That's what I demonstrated in Fig. 1 of Toride et al.(1993). It is not surprising that Cf can be negative when the direction of solute flux is opposite to the direction of the axis.

Regards,

Nobuo

dburnell · Post by **dburnell** » Tue Nov 20, 2012 7:20 pm

Hi Nobuo,

Thanks. I am having trouble finding your paper and want to cite it. If possible, please send me a pdf of the paper. My email address is: Dan.Burnell@Tetratech.com.

One last question. As part of an upcoming paper, I did a simple numerical continuous time random walk (CTRW) simulation for a continous 1-D point source with uniform v, dispersion, and first-order decay at steady-state. I found that the infinite domain solution (transformed to flux-averaged concentration) matched this numerical simulation very closely for all x. On the other hand, the semi-infinite first and third type solutions were close for x>0 but did over- and under-esimate, respectively, the numerical model results. Is it surprising that that the infinite domain solution is a better match (it seems that its boundary condtion is less restrictive on mass transfer in the upgradient direction)?

Thanks again for your time and thoughts!
Regards,
Dan

dburnell · Post by **dburnell** » Tue Nov 20, 2012 8:18 pm

Rien,

Jim Mercer, who is my office, said to say hi. He also wanted to know if you have read his novel.

Regards,
Dan

ntoride · Post by **ntoride** » Wed Nov 21, 2012 1:13 am

Dan,
I probably do not understand well about your calculations. Proper descriptions of the boundary condition have been discussed for long time. I also send you a Rien's comment (SSSAJ, 991-993, 1994). Please look at some of old literatures such as Wehner and Wilhelm (Chem. Eng. Sci., 89-93, 1956) when you have time. These literatures will help you to understand your findings. Best regards,
Nobuo