Fluids (Through an IV) Question

[boneh3ad]

Yes of course (that is well understood), except for the issue of the OP which, as I and others suggested, relates to the T connector; and I gave a reference to a practical application of that phenomenon by Tesla.

And I fully agree with that assessment. That is very likely the cause of a decrease in flow rate in the OP's case rather than seeing no change.

 

[Andy Resnick]

There's a lot in your post I don't understand- maybe you can help walk me through it:

Poiseuille flow, while the correct flow for this situation, is not needed to solve this problem. You don't need to know the velocity profile at a given point in the tubing or needle to know the flow rate, adn that is what Poiseuille flow gives you.

I agree Poisueille flow may not be the only way to solve the problem. I also agree that one does need velocity to calculate a flow rate: for example, Q = \frac{\pi \mu R}{2 \rho}Re, where Re is the Reynolds number.

If you hold velocity constant (and density is obviously constant), doubling D quadruples the mass flow rate. After all, the formula for area only depends on D^2, so doubling D gives you a factor of 4, not 16.

I don't see this. The elementary result, Q = \frac{\pi R^{4}}{8 \mu}\nabla P clearly has a 4-th power dependence on R.

This is, in fact, one of the pressures that can be used to answer this problem. It is dependent on the velocity of the fluid at that point plus the static pressure.

I don't follow this either- pressure, which is part of the overall stress tensor, depends on the velocity *gradient*, not the velocity. Clearly, uniform motion of a fluid should not alter the internal pressure of the fluid (in the region where v is constant). The results of measurements in any inertial frame are the same.

I promise you, Bernoulli's equation absolutely applies to get that and is used all the time.

I think Bernoulli's equation could probably be used as well.

So, what does this simplified system mean? It means you can ignore the velocity going through the system. And only worry about the pressure from the reservoir and the exit pressure. As you stated, the exit pressure is fixed. The pressure of the reservoir, however, is entirely determined by two things: height of the reservoir and atmospheric and other external pressures. It has nothing to do with the amount of fluid in the bag or the number of bags hooked up. It only depends on the highest point the fluid reaches above the exit. In other words, the only way to increase the pressure without changing the diameter of the tubes or the needle is to either lift the bag higher or squeeze it (either with your hands or an external pressure of some kind, it doesn't matter).

Let me make sure I understand- we are talking about the pressure at a specific point- the needle inlet? Because I can change the column height of fluid by adding or removing fluid to/from the bag- that should not be in dispute, either.

The combined mass flow allowed by the stopcocks was simply lower than the maximum allowed by the tubes,

I don't understand what you mean by "maximum (flow) allowed by the tubes". Is there some upper limit? What sets this limit? The flow rate through a tube doesn't admit an absolute maximum- if I double the pressure gradient, I double the flow rate. There is some limit when turbulent flow is reached (say, Re = 4000 and higher), but that's not what we are talking about, AFAIK.

You can show that the force on a differential element is:
dp = -\rho g \; dy
If you integrate this from the fluid element of interest (the exit plane in our case) to the very top surface of the fluid in the reservoir (meaning the heighest point of fluid in the control volume), you get:
\Delta p = -\rho g \Delta h

Ah- I think this is the problem. The pressure at both the top surface and the bottom surface is atmospheric pressure. This causes a lot of confusion- the pressure at the exit plane of the needle is atmospheric pressure. The pressure at the top of the reservoir is also atmospheric pressure. So what we really have is the pressure from the top of the reservoir to the needle inlet is \Delta p = -\rho g \Delta h, which 'h' taken as negative (since the pressure at the needle is *higher* than the pressure at the top free surface), and the pressure (over atmospheric) along the needle is given by \Delta p = \rho g \Delta h/l * x, where 'x' is the distance along the needle (and varies from 0 to l, the length of the needle)- I may have reverse the sense of direction within the needle.

Of course, this is a hydrostatics example. For this problem, we need a moving fluid. Enter Bernoulli's equation, which has the same form as the hydrostatic equation, only including motion terms. If we look at the same two points at the top of the reservoir where the velocity is zero and at the exit of the needle

You lost me- Why is the velocity zero? Fluid is draining from the reservoir- the top surface is moving down.

 

[boneh3ad]

I agree Poisueille flow may not be the only way to solve the problem. I also agree that one does need velocity to calculate a flow rate: for example, Q = \frac{\pi \mu R}{2 \rho}Re, where Re is the Reynolds number.

Well, you can use a Reynolds number but you end up using it for it's velocity component so to speak. You need some form of velocity to get the flow rate. The difference is that you don't need a velocity profile. You can simply use the average velocity over the entire exit area to get mass flow rate. After all, the average velocity, by definition, could be obtained from integrating that velocity profile given by Poiseuille flow. In fact, in integrating the Poiseuille solution across the exit area, you would end up with an average velocity term if you so chose to isolate it.

The important takeaway here is that you use a Poiseuille flow solution to get the velocity profile, which you don't need. The volumetric flow rate that is quoted on Wikipedia is just one of the takeaways from integrating the velocity profile across a plane of interest.

I don't see this. The elementary result, Q = \frac{\pi R^{4}}{8 \mu}\nabla P clearly has a 4-th power dependence on R.

The problem is that Q has an R^2 term in it as well. At its most basic for incompressible flow, the mass flow rate can be written as \dot{m} = \rho v A. In this case, v \propto \Delta p and A \propto D^2. That means that for a given \Delta p, you are left with \dot{m} \propto D^2.

I don't follow this either- pressure, which is part of the overall stress tensor, depends on the velocity *gradient*, not the velocity. Clearly, uniform motion of a fluid should not alter the internal pressure of the fluid (in the region where v is constant). The results of measurements in any inertial frame are the same.

The stress tensor and pressure are not synonymous. The stress tensor does have a dependence on the velocity gradient. The pressure, while a component of the stress tensor, does not. The stress tensor technically speaking is defined as:

\boldsymbol{\tau} = -\nabla p \;\mathbb{I} + \mu \nabla^2 \vec{v}

The pressure term shows up as a gradient, and has nothing to do with the velocity gradient and everything to do with externally enforced pressure gradients or hydrostatic pressure. It has no explicit dependence on velocity.

Let me make sure I understand- we are talking about the pressure at a specific point- the needle inlet? Because I can change the column height of fluid by adding or removing fluid to/from the bag- that should not be in dispute, either.

You absolutely can add fluid to the bag to change the height. I was taking the point to be the needle outlet, but if you were trying to figure out the inlet properties, you could easily do it that way as well and then use your result to do a similar analysis across the needle. I just skipped a step. The difference would be if there was a height difference between the entrance and exit of the needle (unlikely) or an area difference (possible).

I don't understand what you mean by "maximum (flow) allowed by the tubes". Is there some upper limit? What sets this limit? The flow rate through a tube doesn't admit an absolute maximum- if I double the pressure gradient, I double the flow rate. There is some limit when turbulent flow is reached (say, Re = 4000 and higher), but that's not what we are talking about, AFAIK.

I may have used some odd terminology, as there certainly isn't really a maximum flow rate for an incompressible, inviscid fluid. Of course, for very high velocities, viscosity could counteract any additional pressure and compressibility effects can choke the flow to prevent additional mass flow, but neither of those situations apply here.

What I mean is this: imagine you have a garden hose and you are dripping water through it with a turkey baster. That is essentially what you are doing with the stopcock. Your mass flow rate is dictated by the stopcock, and for the same conditions, the stopcock admits less flow than the tubes would. Your stopcock is throttling your flow, in essence. Based on your results, I infer that even the combined mass flow of both stopcocks is likely not as much as the tube could admit under the given flow conditions. In all likelihood, that means there is a bit of unused space in the tubes when the water is draining from your funnels, or in other words, the entire cross section is not filled with water. You are only getting as much water through that system as the stopcocks will allow, which is apparently less than what the tubes would allow.

You lost me- Why is the velocity zero? Fluid is draining from the reservoir- the top surface is moving down.

Looking at an instant in time, the velocity is zero in the reservoir but nonzero at the exit. Even in general, you take the reservoir velocity to be zero in these cases because it is very tiny compared with the rest of the system. You can look at a handful of books and find examples of this, but it is a reasonable approximation except perhaps when the bag is nearly empty. At that point crazy things happen anyway. The top surface is of course moving down, so when looking over a period of time, you would have to take the \Delta h term as a time varying quantity and you would end up with a differential equation.

If you did take \Delta h to be time varying, you would notice that with two bags, the fluid level falls more slowly, so the pressure would stay marginally higher, but since we are only talking a matter of inches here, it still won't speed the bags up enough to pass the single bag, especially because even if the two bags did drain faster, they would catch up to the height of the single bag and end up draining at half the rate again.

In other words, you would expect that in practice, neglecting the effect of the junction downstream, the two bags would probably drain in something like 1.75 times as long as the single bag rather than a perfect 2. Finding the actual value is nontrivial and would need to be numerically integrated, so I don't really plan to do that since I ought to be finishing this paper that needs to be submitted to AIAA by next week.

 

[Andy Resnick]

I may have used some odd terminology, as there certainly isn't really a maximum flow rate for an incompressible, inviscid fluid.

Aha- this is one major problem. I tried to work the problem using Bernoulli's law, and now I am convinced that Bernoulli's law *cannot* be used to correctly analyze this problem, because Bernoulli's law only holds for inviscid flow- Poiseuille flow fails for inviscid flow. I also understand why the introductory textbook picture of Bernoulli's law is presented the way it is, with gravity perpendicular to the flow.

Bernoulli's law is simply conservation of mechanical energy. Let's compare two control volumes, one at the top of the reservoir (say, a large diameter thin cylinder) and one within the needle, terminating at the outlet (a long slender cylinder). In some time dt the volume dV exits the needle. Each control volume dV is the same (since conservation of mass holds and we assume an incompressible fluid) and has a mass dm. Conservation of mechanical energy gives:

P + 1/2 dm v^2 + dm gy = constant.

replacing dm = \rho \Delta V and v = dl/dt = 1/A dV/dt = Q/A (since dV = A dl, and Q = dV/dt), we get for the upper control volume:

P_{atm} + \frac{1}{2} \rho\Delta V\frac{Q^{2}}{A^{2}_{1}}+\rho\Delta Vgh, where h is the height above the needle outlet, and we assumed that l_1 was so thin that the entire control volume is held at atmospheric pressure.

Writing the expression for the lower volume is not so simple, as we will see- since we want A_1 >> A_2 (the diameter of the reserviour is much larger than the needle), the corresponding height l_2 >> l_1, and so we must be careful about assigning a pressure and potential energy. For now, let's do the obvious linearization:

P_{atm} +\delta P + \frac{1}{2} \rho\Delta V\frac{Q^{2}}{A^{2}_{2}}+\rho\Delta Vg \frac{l_{2}}{2}

and thus Bernoulli's law gives:

\delta P = \rho\Delta V [ \frac{Q^{2}}{2}(\frac{1}{A^{2}_{1}}-\frac{1}{A^{2}_{2}}) + g(h-\frac{l_{2}}{2})]

Now we take some limits: A_1 >> A_2, h >> l_2:

\delta P = \rho\Delta V [ -\frac{Q^{2}}{2}\frac{1}{A^{2}_{2}} + gh)].

From this, we can generate Torricelli's result v = \sqrt{2gh} by setting dP = 0. This corresponds to putting a hole in the side of the container, so there is no pressure difference across the ends of the lower control volume. But, our control volume in the needle is vertical, and so there is a pressure difference between the top and bottom face. Since we can't use the result from Poiseuille flow, it's not entirely obvious what to put for this pressure difference.

The bottom line, Bernoulli's law is not appropriate to model viscous flow. All discussions about "maximum flow capacity", the roles of "flow restrictors" and the like only arise when trying to force inviscid flow equations to handle viscous effects.

 

[boneh3ad]

Aha- this is one major problem. I tried to work the problem using Bernoulli's law, and now I am convinced that Bernoulli's law *cannot* be used to correctly analyze this problem, because Bernoulli's law only holds for inviscid flow- Poiseuille flow fails for inviscid flow. I also understand why the introductory textbook picture of Bernoulli's law is presented the way it is, with gravity perpendicular to the flow.

...

The bottom line, Bernoulli's law is not appropriate to model viscous flow. All discussions about "maximum flow capacity", the roles of "flow restrictors" and the like only arise when trying to force inviscid flow equations to handle viscous effects.

Aha, but this isn't true. Bernoulli's principle can be used on a viscous flow just fine. The kicker is that it doesn't give you the exact velocity profile and needs correction factors (typically called head loss) to correct for viscous losses. As long as you know the limitations, you can successfully use it to analyze a viscous problem. If you would like, I could go look up the correction factor for friction for our system. Otherwise, you can do it. It is called Darcy's friction factor or Colebrook's friction factor, depending on how accurate of a result you want. Colebrook's value is more accurate but is an implicit equation.

Anyway, all that friction loss will do is change the total pressure drop between the bag and the needle, meaning a lower exit velocity. It doesn't change the fundamental dependencies of the solution.

Also, the flow restriction talk has nothing to do with viscous or inviscid flow. It has to do with the stopcock not physically being able to physically pass the same amount of mass through it for a given flow condition. That has nothing to do with viscosity and everything to do with simply the size of the hole. Stopcocks are designed to restrict flow rate. YOur stopcocks are doing their jobs.

 

[Andy Resnick]

Aha, but this isn't true. Bernoulli's principle can be used on a viscous flow just fine.

I'm going to ask for a reference on this one- everything I found was considerably more equivocal:

http://arxiv.org/pdf/0809.1309

http://www.google.com/url?sa=t&sour...sg=AFQjCNF86DJxtd0xu6I5PvBRw8P9cG55Sw&cad=rja

http://www.engineersedge.com/fluid_flow/extended_bernoulli.htm

The second reference has the fluid exiting out of a hole in the side, which makes direct comparison to our situation questionable. The third reference requires empirical input to solve- which begs the question why it is preferred in our simple system.

 

[boneh3ad]

I'm going to ask for a reference on this one- everything I found was considerably more equivocal:

Try any basic fluid mechanics textbook geared towards mechanical engineers. You can change the viscous term into a head loss, which shows up in the form of a length so that it can be directly compared to a change in height. This is done commonly with the Darcy–Weisbach equation for the case of the head loss due to friction in the pipe (or otherwise known as viscous dissipation).

A book I happen to have on hand here that I can cite is "Fundamentals of Fluid Mechanics" by Munson, Young and Okiishi (https://www.amazon.com/dp/0471675822/?tag=pfamazon01-20). Particularly check the section about pipe flow. You essentially get a corrected Bernoulli equation.

Other assumptions necessary, such as fully-developed flow, can also be accounted for in terms of an energy loss term that shows up as a head loss. The same can be said for the T where the IV bags come together, and that is where the slowup comes from in the OP's example, since friction is present in both cases.

Still, none of them would support your original claim that two bags would create twice the mass flow rate. The only way that you can get more flow rate is by raising the bags higher, putting pressure on the bags or increasing the exit diameter of the needle. That goes for the viscous case, the inviscid case, and any other incompressible case you can think of that involves obeying the laws of physics.

 

[edgepflow]

In practice, the Bernoulli Equation is corrected for viscous flow with a "discharge coefficent." Similar to the equation boneh3ad derived earlier, we can express the volumetric flow rate as:

Vdot = Cd * Aorf * sqrt (2gh)

where,

Vdot = volumetric flow rate
Cd = discharge coefficient of opening = actual discharge flow / theoretical discharge flow
Aorf = area of discharge opening
h = fluid level.

As was derived earlier by boneh3ad and Andy Resnick, the surface area of the reservoir does not appear in this expression.

 

[Andy Resnick]

Try any basic fluid mechanics textbook geared towards mechanical engineers.

Ah- I should have realized I was dealing with an engineer. I was able to get oriented with Vennard and Street.

Look up the three-reservoir problem. The flow rates add- the flow out of the nozzle Q_3 is equal to the flow in from both funnels: Q_1 + Q_2 = Q_3.

http://cee.engr.ucdavis.edu/faculty/bombardelli/Three_reservoir_problem.pdf

http://personalpages.manchester.ac.uk/staff/david.d.apsley/hydraulics/threeres.htm

http://excelcalculations.blogspot.com/2011/05/three-reservoir-problem.html

 

[cjl]

Of course the flow rates add. That's simple continuity. That's not the same thing as saying that the flow rates from each reservoir while both are flowing are the same as they would be with only one reservoir supplying the pressure (or, to put it another way, the flow rates are found in a way which is dependent on the overall arrangement - they can't be found independently and then summed). It is also, as has been explained to you extensively, highly dependent on the details of the problem, specifically where the majority of the pressure loss occurs.

(I could also note that you haven't addressed my experimental results yet)

 

[boneh3ad]

Ah- I should have realized I was dealing with an engineer. I was able to get oriented with Vennard and Street.

Look up the three-reservoir problem. The flow rates add- the flow out of the nozzle Q_3 is equal to the flow in from both funnels: Q_1 + Q_2 = Q_3.

http://cee.engr.ucdavis.edu/faculty/bombardelli/Three_reservoir_problem.pdf

http://personalpages.manchester.ac.uk/staff/david.d.apsley/hydraulics/threeres.htm

http://excelcalculations.blogspot.com/2011/05/three-reservoir-problem.html

And we are dealing with a biophysicist. What is your point? Fact: you clearly can't see when you are mistaken and you seem to hold engineers in low regard. Good day.

 

[TrickyDicky]

Marcusl,
Cjl and sams_rhythm, your attitude is deplorable and you should be very thankful that you are not my students.

Hilarious.
One can be dead wrong, it's OK, even being a professor, but being wrong and brag about it ...

BTW, very informative thread, thanks cjl and the rest.

 

[Andy Resnick]

And we are dealing with a biophysicist. What is your point? Fact: you clearly can't see when you are mistaken and you seem to hold engineers in low regard. Good day.

Now you are being intentionally ignornant:

Now hook up two bags that T together and come out of the same sized final orifice with all bags and the exit at the same respective heights. The pressure is still \rho g h and the area is still A, so that mass flow rate is still the same.

Because it is wrong. Adding two bags does not double the flow rate.

Still, none of them would support your original claim that two bags would create twice the mass flow rate.

Meanwhile:

By hooking up multiple bags, you are increasing the pressure drop (using two bags at the same height doubles the driving pressure) and so you increase the flowrate.

Here's the setup:

The drainage times were unchanged. Thus the flowrate was changed

I don't hold engineers in low regard- I hold people who insist on being wrong in low regard:

http://en.wikipedia.org/wiki/Dunning–Kruger_effect

Shameful.

 

[boneh3ad]

Take a peek in the mirror then. Years of experimental evidence contradicts your claims here. Years of theory by actual fluid dynamicists contradicts your claims here. There are a handful of fluid dynamicists IN THIS thread telling you that you are incorrect. Which of us is the one who is shameful?

You have now called me unskilled and lacking the cognitive capacity to realize I am wrong as a result, yet my claims are borne out by experiments, theory, and other fluid dynamicists everywhere. Which of us seems to fit your accusation better? Or do you claim that EVERYONE is less competent than you, even in fields in which you aren't familiar? You, sir, are delusional.

Look, nobody is perfect. I have certainly had a couple holes in my argument throughout this thread that I have tried to patch up/clarify as it has gone on. After all, that is why I come here: to work on my own ability to explain these concept and to converse with other people who can share another point of view or who are looking to learn.

Earlier you said you were glad some of us weren't your students. Well, I believe I speak for the lot of us when I say that we are glad as well, especially since I have no interest in biophysics and you have no clue in fluid dynamics.

 

[Andy Resnick]

Take a peek in the mirror then. Years of experimental evidence contradicts your claims here. Years of theory by actual fluid dynamicists contradicts your claims here. There are a handful of fluid dynamicists IN THIS thread telling you that you are incorrect. Which of us is the one who is shameful?

On the contrary- the evidence supports my statements. Or rather, my experiment is in agreement with everyone else-

http://www.engr.scu.edu/~emaurer/classes/ceng141_hydraulics/handouts/Hwk06_plumbing_3_reservoir.pdf

http://www.efm.leeds.ac.uk/CIVE/CIVE2400/Handouts/threep.pdf

 

[boneh3ad]

I feel the need to point out that the first link you provided there uses my method of the modified Bernoulli equation. It has head loss terms to account for inviscid effects such as viscous dissipation at the wall, bends in the pipes, mixing, etc.

The second source you just linked to uses my method again, even referring to the same equation I made reference to earlier in Darcy's equation for friction loss in a pipe. Now, if you are referencing the bottom where it says Q_a + Q_b = Q_c then of course that holds for an incompressible fluid, as the mass of the system must be conserved. The thing is, if you closed off one of the branches (a or b) then the other one would show an increase in Q since the second branch is no longer contributing.

The important thing though is that both of these problems are analyzed exactly as I have described in earlier posts. Bernoulli's equation, which serves as the energy equation in this case, can be used if you either neglect viscosity (a useful approximation for many, many problems that gives a physically meaningful answer, even if it isn't the exact solution) or correct for it in the form of a head loss.

 

[berkeman]

Thread closed for Moderation...

 

[berkeman]

This is still being discussed at length in the Mentor forums, both on a technical level and on a PF rules level. Thanks for your patience.