Flow Rate: Example

by Jared Rovny

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    00:01 Let’s take an example of this. Let’s suppose you have an arteriole and has a radius of 30 micrometers which we often call microns, a radius of 30 microns and a flow velocity of 1 centimeter per second.

    00:12 Then it splits as it goes into different parts of your muscles and delivers oxygen into 1000 different capillaries, each one with a much smaller radius of 10 microns each. The question is what would be the flow velocity in the capillaries and what is the ratio of the flow rate through one such capillary to flow through the arteriole? Hopefully, you've tried this yourself. I definitely recommend giving this a pause and seeing if you can come up with an answer. Let’s go ahead and jump in and see what this looks like. If you have such an arteriole which has a much bigger radius and you have the fluid is flowing through this channel first, where we’ve said again that this has a radius of 30 microns.

    00:59 Then this is going to split into many, many, many different channels. Each one of these channels is going to be much, much smaller. These are all your capillaries. So, you have your blood is flowing into many, many smaller, different channels as it goes. So each one of this is very small and in fact has a radius that we said is only 10 microns. We can find the cross sectional area since each one of these is actually a cylindrical area and we know the radius just by looking at this circle, this cross sectional area of your cylinder. So the area of this cross section is π times the radius of the vessel, whether it's the arteriole or your capillaries, squared. So, this is going to be the area that we find.

    01:43 Again, we said that flow rate, Q is conserved. So, Q1 flowing through your arteriole has got to be equal to Q2 which is the total flow rate through all of these capillaries together. So, this is equal to Q2.

    01:57 We can go ahead and write this out. We have A1V1 is equal to A2V2. Now, we just need to find these areas and see if we can find the velocities based on that. What we're interested in is V2 for this first part of the problem. What’s the velocity going through the capillaries? So, V2 will just be A1 over A2 times V1 as a ratio. So, all we need now is to find the ratio of this area since we already know this velocity. Let’s go ahead and plug this in. What we see is that V2 is equal to A1 which is just the cross sectional area of your arteriole, πr1 squared over A2 which is the cross sectional area of your capillary. This is πr2 squared. Now, what we have to do is multiply that by V1.

    02:51 We can cancel these π’s. Now, we have r1 over r2, so we’ll have to take this as a ratio, squared.

    02:59 This is just a way of rewriting it because it might be easier to take the ratio before we do the squaring rather than afterwards. In here, we have the ratio of the radii, the radii of your arteriole and capillaries times the velocity. So, let’s do plug in our numbers and see what we get.

    03:14 The radius of your arteriole is 30 microns. Now, here’s the trick. It might be tempting just to say that the radius that we’re going to use here is just a 10-micron radius. The problem is that this area here is not just the cross sectional area of 1 capillary. We, in fact, have many, many, many capillaries.

    03:36 In fact, we’re told that we have 1000 of this. So, this flow rate equation has a caveat which is that the flow rate through the entire system has to be conserved. So the flow, Q1 did not just go through one capillary. It went through all of your capillaries.

    03:51 So, when we’re analyzing the flow Q2, we’re not just talking about the flow through one of this but the flow through all of these capillaries together. That’s going to be a much greater, total cross sectional area. So, there’s one thing that we actually have to do with this equation down here which is to say that A2 is not referring to the cross sectional area of just one capillary.

    04:09 It’s actually referring to the cross sectional area of all of these capillaries together.

    04:13 So, let’s make that edit and make sure that we have this physically correct. We have the velocity 2 is equal to A1 over the total A2 which is all of these capillaries together. This is going to change our answer in a very important way. So, this is divided by 1000 now. Now, we still have this radius r2.

    04:30 That radius is 10 microns. Let's plug that in. This is all squared times V1 and then divided by 1000.

    04:41 So, let’s plug in these numbers. 30 microns, the units cancel, the 0’s cancel. So, we just have 3 squared.

    04:47 We have that V2 is equal to 3 squared which is 9 over 1000 times V1. We’re given in this problem that the velocity going through the arterioles is 1 centimeter per second. So now, we can see that the flow through the capillaries is only 9 over 1000 centimeters per second.

    05:15 So, we see that the speed or the velocity at which the fluid is flowing through your capillaries is actually much, much slower than the speed at which the blood is flowing through the arterioles.

    05:25 This is very important. This is a clever mechanical trick by your body to split an arteriole into so many different capillaries because not only is it able to reach many, many areas and a greater volume by splitting into so many, but by having a total cross sectional area from so many capillaries would be so great. You can greatly slow down the speed of your fluid and still have a continuous flow rate. The fact that this speed is so slow allows the blood to transition and transfer oxygen and take nutrients and give nutrients and do everything it needs to do at a particular blood site. Then when it comes back to a particular vein which is a much bigger cross sectional area, it can flow again much quicker to go get that oxygen and come back and repeat the process. We have one more part in this problem which is that we would like to know the ratio of the flow rate in a capillary to that in the arterioles. So, let’s quickly do that.

    06:15 We would like to know what the ratio of the flow rates is. We could write this as Q1 over Q2 or Q2 over Q1, if we wanted to find the ratio of that in the capillaries to that of the arterioles.

    06:33 Because the flow rate is conserved, this Q2 and Q1 are still the same. So, the ratio of the flow rate is still just 1. These are no different from each other. So, this would be sort of like a trick question.

    06:46 They might ask you to find some velocity. They might ask you to see what the difference in the velocity is. Then they might ask you to find the ratio of the flow rate. You might be tricked into thinking, "Oh the flow is much slower through the capillaries." It is true that the flow is slower through the capillaries. But the flow rate, the total amount of fluid that’s moving is still the same whether you’re in the capillaries or the arterioles because it's a closed system.

    07:08 Again, we have to have conservation of flow rate through any given closed system.

    About the Lecture

    The lecture Flow Rate: Example by Jared Rovny is from the course Fluids.

    Included Quiz Questions

    1. 3 m/s
    2. 6 m/s
    3. 12 m/s
    4. 2 m/s
    5. 1.5 m/s
    1. Q₁ = Q₂ = 0.012 m³/s
    2. Q₁ = Q₂ = 1.2 m³/s
    3. Q₁ = 1.2 m³/s , Q₂ = 0.3 m³/s
    4. Q₁ = 0.012 m³/s , Q₂ = 0.003 m³/s
    5. Q₁ = Q₂ = 1200 cm³/s

    Author of lecture Flow Rate: Example

     Jared Rovny

    Jared Rovny

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    Helpful lecture with clear guidance
    By Jakob M. on 01. March 2017 for Flow Rate: Example

    Very good lecture that covers one of the core principles of the vascular system through math, and gives a headsup for what might come to the exam. I would, though, like to see more than one example solved.