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I'm trying to educate myself on modelling solute flows through pipe networks.

This is a follow up of my previous post here

$$\frac{\partial C}{\partial t} = - v\frac{\partial C}{\partial x}$$

While modelling flow through pipe junctions, I understand that the junction is assumed to have complete mixing, negligible volume, and algebraic equations are written for mass conservation.

However, I recently found an illustration in a notes posted here(the same figure is provided below), the junctions are displayed to have spherical volume. Is it appropriate to consider that the pipe junctions have spherical volumes? I've been asked to do so for discretizing and solving the solute transport equation.

enter image description here

This really confuses me. On one hand, the velocity of fluid flowing in each branch of the pipe is computed with the assumption that there is no accumulation at the pipe junctions using the continuity equation (Inflow rate is equal to outflow rate). Wouldn't the velocities obtained by solving the continuity equation be wrong if the nodes are assumed to have spherical volume?

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  • $\begingroup$ I think from "spherical control volume" they mean junction is isotropic completely. $\endgroup$ – Alone Programmer Dec 11 '19 at 17:17
  • $\begingroup$ @AloneProgrammer Thank you. But I am advised to consider a spherical volume at the junction and write a differential equation at the junction too. This appears to be incorrect to me. I'm not sure if an algebraic equation has to be used at the junction nodes or a differential equation has to be written( which is not done in any of the literature). $\endgroup$ – Natasha Dec 11 '19 at 17:21
  • $\begingroup$ Obviously from the picture it seems the variation of concentration in the junction itself might be considerable that lead the authors to consider a spherical control volume at the junction. When you ignore the volume of the junction and consider it as a point and write the mass balance equation as an algebraic equation of balance of incoming and outcoming fluxes, you make an assumption that the concentration gradient at the junction (remember any real junction is not a point and certainly it has a volume) is negligible, which might not be the case in your application. $\endgroup$ – Alone Programmer Dec 11 '19 at 17:41
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    $\begingroup$ In fact, by considering a volume for junction you assume that junction does not exchange incoming and outcoming fluxes immediately and also it needs sometimes to transport the mass based on diffusion or convection. For really small junctions in comparison to the volumes of the pipes, this junction delay time might be extremely small and that's the reason why people assume the junction is just a volumeless point. But, if the junction has a comparable volume to the pipes themselves as you show in your picture, you need to solve convection-diffusion equation even at the junction. $\endgroup$ – Alone Programmer Dec 11 '19 at 17:44
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    $\begingroup$ Yes but that's not the case here cause junction has a volume right now and it takes sometimes to transport mass or momentum. For momentum transport it might be less important cause usually $Sc = \frac{\nu}{D}$ Schmidt number is in the order of 1000 for typical fluids and it means momentum is transported 1000 times faster than mass, so you may ignore the volume of the junction for the velocity safely but not for the mass transport. $\endgroup$ – Alone Programmer Dec 11 '19 at 17:48
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Obviously from the picture it seems the variation of concentration in the junction itself might be considerable that lead the authors to consider a spherical control volume at the junction. When you ignore the volume of the junction and consider it as a point and write the mass balance equation as an algebraic equation of balance of incoming and outcoming fluxes, you make an assumption that the concentration gradient at the junction (remember any real junction is not a point and certainly it has a volume) is negligible, which might not be the case in your application.

In fact, by considering a volume for junction you assume that junction does not exchange incoming and outcoming fluxes immediately and also it needs sometimes to transport the mass based on diffusion or convection. For really small junctions in comparison to the volumes of the pipes, this junction delay time might be extremely small and that's the reason why people assume the junction is just a volumeless point. But, if the junction has a comparable volume to the pipes themselves as you show in your picture, you need to solve convection-diffusion equation even at the junction.

Yes but that's not the case here cause junction has a volume right now and it takes sometimes to transport mass or momentum. For momentum transport it might be less important cause usually $Sc=\frac{\nu}{D}$ Schmidt number is in the order of 1000 for typical fluids and it means momentum is transported 1000 times faster than mass, so you may ignore the volume of the junction for the velocity safely but not for the mass transport.

Update:

Based on OP's actual application for capillary blood vessels, I think it's pretty straightforward to define spherical volume of junction by considering the Maximum Inscribed Radius of all branches as is shown here to give you an idea of what would be spherical junction in case of even patient-specific blood vessel geometries (specifically look at the sphere in the right image at the junction of those three branches):

enter image description here

Picture reference: VMTK, http://www.vmtk.org/tutorials/BranchSplitting.html

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  • $\begingroup$ Thanks a ton!. May I have the link to the reference from which the above image has been obtained. I'm also interested in understanding how the discretization scheme would look like at the junction that has a volume. $\endgroup$ – Natasha Dec 11 '19 at 18:16
  • $\begingroup$ On a side note: Is this possible to obtain patient-specific geometries like this in any of the public repositories? I've been looking for data of this type. I'd be super happy if you were my mentor! $\endgroup$ – Natasha Dec 11 '19 at 18:22
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    $\begingroup$ @Natasha I added the reference for the picture. I'm not sure about discretization of the junction. It depends on your numerical scheme. For example, FEM may work good with tetrahedral meshes but it pretty much depends on your computational framework. This picture is not a capillary artery by the way, but it depends on which part of the body you want to obtain patient-specific geometries. Now that you clarified that you want to solve this problem for blood arteries it's good to have a look at this repository: github.com/INSIGNEO/openBF $\endgroup$ – Alone Programmer Dec 11 '19 at 18:59
  • $\begingroup$ Thanks a lot for the reference. I had a look at openBF. From what I understand, openBF is a really nice 1D model used for simulating blood flows in vascular networks. Unfortunately, I couldn't find studies that have used openBF for simulating solute flows. $\endgroup$ – Natasha Dec 12 '19 at 5:14
  • $\begingroup$ @Natasha Yes, openBF is only for blood flow simulation through vascular networks, but it just can give you an idea what would be the procedure to develop a 1D network solver. Also, please look at this repository: github.com/JihoYang/Connolly which combined blood flow and drug concentration transport through complex vascular networks, which might be more interesting for you. $\endgroup$ – Alone Programmer Dec 12 '19 at 15:06

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