2D advection-dispersion velocity component for fluid flow in a pipe

[maistral]

TL;DR Summary

Assuming the velocity vector for the radial direction is not zero, how do I get it? And other questions.

Pardon if the answers to my questions are obvious, because as usual I am trying to decipher everything on my own (as the material has not been taught to us quite well; then again it's graduate school). I just need someone to reassure me that I am understanding this correctly.

Say for example I have a pipe where the 2D advection dispersion equation applies on the flowing fluid (assume tangential/angular gradients are negligible), I will have the quantity u_r for the velocity along the radial direction and u_z for the axial direction.

1. Am I correct to assume that the 'true' velocity would be the resultant of these two? ergo, u = √(u_r² + u_z²), where u is the 'true entrance velocity' of the fluid in the pipe (and that u = u_z because it is a general assumption that u_r is zero since this is a pipe, and that radial velocities are near inexistent?)

2. Assuming that there are density differences (due to temperature), obviously there will be differences to the point velocities as velocity = volumetric rate / area; and that volumetric rate and density are inversely related. If say for example, u_z is zero, I am assuming that ∂u_z/∂z is not necessarily zero due to these density differences in different points along the fluid region (change in densities will cause change in volumetric rate). Is this correct?

3. Also, if I want to calculate u_r from the point volumetric rate, then it's u_r = volumetric rate / radial area, and I am assuming the radial area = 2πrL?

4. I know this is so undergraduate and stupid (pardon me, lol). I have a quantity D_i for diffusivity coefficient which is a function of flow rate F_i of two components, temperature T, and pressure P, and that D_i returns a vector containing the diffusivities of two components. If I intend to calculate ∂D_i/∂z, I can use chain rule. I am assuming this is correct, lol.

So if I intend to calculate ∂D_i/∂z, I can calculate (∂D_i/∂F_i) (∂F_i/∂z), which would probably be inefficient since I have to keep calling the diffusion function one by one to calculate the derivative for the first component and the second component. I thought of just chaining it to ∂D_i/∂T such that (∂D_i/∂T) (∂T/∂z) so that the returning vector would automatically be derivatives of the diffusion coefficient of each component. I am actually thinking this is allowable since I did not exactly break any mathematical law, lol. It's just that I want to make sure because manipulating my program with respect to T is so taxing.

Thank you so much in advance.

 

[Chestermiller]

Summary: Assuming the velocity vector for the radial direction is not zero, how do I get it? And other questions.

Pardon if the answers to my questions are obvious, because as usual I am trying to decipher everything on my own (as the material has not been taught to us quite well; then again it's graduate school). I just need someone to reassure me that I am understanding this correctly.

Say for example I have a pipe where the 2D advection dispersion equation applies on the flowing fluid (assume tangential/angular gradients are negligible), I will have the quantity u_r for the velocity along the radial direction and u_z for the axial direction.

1. Am I correct to assume that the 'true' velocity would be the resultant of these two? ergo, u = √(u_r² + u_z²), where u is the 'true entrance velocity' of the fluid in the pipe (and that u = u_z because it is a general assumption that u_r is zero since this is a pipe, and that radial velocities are near inexistent?)

If you are talking about flow in the entrance region of the pipe, then, by continuity, the radial velocity is non-zero.

2. Assuming that there are density differences (due to temperature), obviously there will be differences to the point velocities as velocity = volumetric rate / area; and that volumetric rate and density are inversely related. If say for example, u_z is zero, I am assuming that ∂u_z/∂z is not necessarily zero due to these density differences in different points along the fluid region (change in densities will cause change in volumetric rate). Is this correct?

I don't understand this question.

3. Also, if I want to calculate u_r from the point volumetric rate, then it's u_r = volumetric rate / radial area, and I am assuming the radial area = 2πrL?[/quoe]
I don't understand this question either.

4. I know this is so undergraduate and stupid (pardon me, lol). I have a quantity D_i for diffusivity coefficient which is a function of flow rate F_i of two components, temperature T, and pressure P, and that D_i returns a vector containing the diffusivities of two components. If I intend to calculate ∂D_i/∂z, I can use chain rule. I am assuming this is correct, lol.

You're saying that diffusivity is a function of concentration, temperature, and pressure, and these parameters are changing? If this is what you are asking, then, yes, you can use the chain rule.

So if I intend to calculate ∂D_i/∂z, I can calculate (∂D_i/∂F_i) (∂F_i/∂z), which would probably be inefficient since I have to keep calling the diffusion function one by one to calculate the derivative for the first component and the second component. I thought of just chaining it to ∂D_i/∂T such that (∂D_i/∂T) (∂T/∂z) so that the returning vector would automatically be derivatives of the diffusion coefficient of each component. I am actually thinking this is allowable since I did not exactly break any mathematical law, lol. It's just that I want to make sure because manipulating my program with respect to T is so taxing.

Thank you so much in advance.

You are trying to solve a problem in which diffusivity of a species is changing with spatial position numerically? Why are you not keeping the diffusivity inside the spatial gradient along with the concentration derivative? This is a better way to do the spatial discretization. And you won't have to evaluate partial derivatives of the diffusivity with respect to temperature, pressure, and composition.

 

[maistral]

If you are talking about flow in the entrance region of the pipe, then, by continuity, the radial velocity is non-zero.

May I ask how to solve for this? Say I have an entrance velocity u of 10 in SI units. How do I know who is u_z and who is u_r?

I don't understand this question.

I was going to express the point velocity u_z or u_r in terms of volumetric rate. But since volumetric rate changes with temperature, and temperature changes with position (due to heat exchange or generation), then volumetric rate must change with position, then in turn, point velocity changing with respect to position.

If I initially assumed that my u_r is insignificant (zero), it is still possible to get a value for ∂u_r/∂r because of this, yes?

I don't understand this question either.

If I have a volumetric rate of 10 in SI, and assuming the volumetric rate changes because of the position (same issue I had with the previous question), I wanted to express u_r as a function of the point volumetric rate. Which area do I use, 2πrL?

You are trying to solve a problem in which diffusivity of a species is changing with spatial position numerically? Why are you not keeping the diffusivity inside the spatial gradient along with the concentration derivative? This is a better way to do the spatial discretization. And you won't have to evaluate partial derivatives of the diffusivity with respect to temperature, pressure, and composition.

I was thinking about this long and hard, to the point that roommates were reprimanding me that I looked like someone constipating from too much thinking about this, hahaha. I just got on instead of overthinking and expanded the flux derivatives. I should have finite-differenced fluxes instead. Lol, thanks for answering a question that boggled me for a week. Next time I would remember this, thank you very very much.Also, I have this last... particular question. At the entrance of my pipe, temperature, total volumetric rate (say, 5 L/s), and pressure is constant. Say I have 10 nodes assigned at the pipe entrance. I have a total molar rate entering the pipe. (say 3 mol/s). I wanted the concentration at the entrance to solve my diffusion equation. What should I put at the top nodes?

• Do I divide the 5L/s by 10, since there are 10 nodes; then I would divide the 3mol/s by 10 as well, since there are 10 nodes; then I will get the concentration from the quotient of the two (3÷5 mol/L to all ten nodes)?
• Or do I hold the 5 L/s constant to all of the nodes, then chop the 3 mol/s to each node (0.3÷5 mol/L to all ten nodes)?
• Or do I hold the molar rate constant, then divide the 5L/s to all 10 nodes (3÷0.5 mol/L to all ten nodes)?
• Or do I not divide anything by anything (3÷5 mol/L to all ten nodes)?

If it is either the first or the last, please do tell me if I have to divide the volumetric rate or not. I'm trying to understand the concepts of things.

Thank you, very very much.

 

[Chestermiller]

May I ask how to solve for this? Say I have an entrance velocity u of 10 in SI units. How do I know who is u_z and who is u_r?

I don't fully understand what you are asking here, but, if you are interested in getting a start in your thinking about entrance flows, study the Blasius solution to flow past a flat plate, which handles velocity components in both the x and y directions, even though the main component of velocity is in the x direction. Entrance flow in a pipe begins like the Blasius solution, since the boundary layer is thin compared to the pipe radius, and thus, the curvature can be neglected.

I was going to express the point velocity u_z or u_r in terms of volumetric rate. But since volumetric rate changes with temperature, and temperature changes with position (due to heat exchange or generation), then volumetric rate must change with position, then in turn, point velocity changing with respect to position.

If I initially assumed that my u_r is insignificant (zero), it is still possible to get a value for ∂u_r/∂r because of this, yes?

I don't know what you are trying to do here. Sorry..

If I have a volumetric rate of 10 in SI, and assuming the volumetric rate changes because of the position (same issue I had with the previous question), I wanted to express u_r as a function of the point volumetric rate. Which area do I use, 2πrL?

Again, I'm clueless about what you are getting at.

Also, I have this last... particular question. At the entrance of my pipe, temperature, total volumetric rate (say, 5 L/s), and pressure is constant. Say I have 10 nodes assigned at the pipe entrance. I have a total molar rate entering the pipe. (say 3 mol/s). I wanted the concentration at the entrance to solve my diffusion equation. What should I put at the top nodes?

• Do I divide the 5L/s by 10, since there are 10 nodes; then I would divide the 3mol/s by 10 as well, since there are 10 nodes; then I will get the concentration from the quotient of the two (3÷5 mol/L to all ten nodes)?
• Or do I hold the 5 L/s constant to all of the nodes, then chop the 3 mol/s to each node (0.3÷5 mol/L to all ten nodes)?
• Or do I hold the molar rate constant, then divide the 5L/s to all 10 nodes (3÷0.5 mol/L to all ten nodes)?
• Or do I not divide anything by anything (3÷5 mol/L to all ten nodes)?

If it is either the first or the last, please do tell me if I have to divide the volumetric rate or not. I'm trying to understand the concepts of things.

If I understand you correctly, I would choose the first.