TBR CBT 2 question 20

[justhanging]

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As a fluid flows through a tube whose radius steadily decreases from 2.0 to 1.0 cm, what change can be expected if the indicated factor remains constant?


A.
If flow rate is constant, then ∆P must increase
B.
If flow rate is constant, then ∆P must decrease

Using the continuity formula and bernoli's equation you can show (unless am wrong) that the pressure at 2.0 cm part is going to bigger than at the 1cm section. The passage gives this formula:

flow rate = ∆P * (R^4/c)

the c in the above equation is just a bunch of constants that don't matter. Why is the answer A? I feel like it should be B, I think it has to do with what is meant by delta P that is confusing because it seems obvious that the pressure would be larger at the 2cm portion.

 

[BenZq]

This question involves using Poiseuille's Law:

If Q is held constant and r is decreased, then P must increase to maintain Q.

I see your point about Bernoulli's equation, so i can't really say why your wrong. Maybe it has to do with a change in the elevation head or maybe your not supposed to use it in conjunction with the continuity equation. Either way, this type of problem is meant to be attacked with Poiseuille's Law.

 

[ilovemcat]

This question involves using Poiseuille's Law:

If Q is held constant and r is decreased, then P must increase to maintain Q.

I see your point about Bernoulli's equation, so i can't really say why your wrong. Maybe it has to do with a change in the elevation head or maybe your not supposed to use it in conjunction with the continuity equation. Either way, this type of problem is meant to be attacked with Poiseuille's Law.

Here's the same equation written in a different way:

deltaP = QR (where Q is flow rate under non-ideal conditions and R is resistance). Note: The "R" term or Resistance is just a combination of variables in Poiseulli's Law; R = 8nl/pi r^4.

As the radius gets smaller and smaller, you probably know from experience that resistance will increase (you can verify this with Poiseulli's Law). Since we're told the flow rate "Q" will remain constant, answering the question becomes very easy. Change in Pressure is proportional to Q . If Resistance (R) increases but Flow Rate (Q) remains the same, then change in pressure must increase.

In the absence of resistance, flow remains constant (Q=Av)
In the presence of resistance, flow rate decreases (Poiseuille's Law); To keep flow rate constant, change in pressure must increase.

Flow Rate (Q) = Change in Pressure (deltaP) / Resistance (R)

Using the continuity formula and bernoli's equation you can show (unless am wrong) that the pressure at 2.0 cm part is going to bigger than at the 1cm section.

The continuity formula - Q = Av and Bernoulli's Equation both apply to Ideal Fluid Flow.

P1+ rhogy1 + (1/2rhov^2)1 = P2 + rhogy2 + (1/2rhov^2)2

Let's say fluid is flowing from a 2cm section to a 1cm section:

- Radius Decreases (therefore A decreases); Because Q is constant, a decrease in A means an increase in velocity. (Q = Av)
- Now let's assume the tube's are connected at equal heights. This way we can throw out the "rhogy" factor in Bernoulli's equation. The equation simplifies to this:

P1+ (1/2rhov^2)1 = P2 + (1/2rhov^2)2

As stated before, a decrease in radius along the tube would result in an increase in velocity (from 2cm --> 1cm). In order for Bernoulli's equation (above) to hold true, there must be a decrease in pressure at that point.

For Bernoulli's equation the terms on each side of the equation can be viewed as "Energy." Resistance takes away some of that energy which is why we don't use Bernoulli's equation for non-ideal fluid flow.

 

[justhanging]

delP = QR
In the presence of resistance, flow rate decreases (Poiseuille's Law); To keep flow rate constant, change in pressure must increase.

What exactly do you mean by 'change of pressure'? because the way I see it now is that the pressure on the 1cm end is still smaller than at the 2cm but like how you said, with resistance, the difference in the two pressures must increase in order to keep flow rate constant. Now that makes sense. So I was right about the pressure being lower at the 1cm end but difference in pressures must increase in order to maintain constant flow.

 

[ilovemcat]

What exactly do you mean by 'change of pressure'? because the way I see it now is that the pressure on the 1cm end is still smaller than at the 2cm but like how you said, with resistance, the difference in the two pressures must increase in order to keep flow rate constant. Now that makes sense. So I was right about the pressure being lower at the 1cm end but difference in pressures must increase in order to maintain constant flow.

By change in pressure - I mean a pressure gradient (ie. a region of High Pressure ---> Low Pressure). Let's say you kept this pressure gradient constant but increased the resistance somehow. According to Poiseulli's Law (delP = QR), flow rate would have to decrease. Therefore, to maintain constant fluid flow we need to increase the pressure gradient.

By the way, I adjusted my previous post a bit to hopefully make things more clear.

 

[BerkReviewTeach]

I think one of the most common mistakes people make with this topic is not seeing the difference between Poiseuille's Law and Bernoulli's equation.

Keep the differences as simple as possible. Think of deltaP in Poiseuille's Law as end-to-end pressure difference. The bigger it is, the faster the molecules travel, which causes the flow rate to increase. The fluid always flows from the end with higher pressure to the end with lower pressure.

Bernoulli's principle can be thought of as the pressure against the inside of the pipe's walls (not the end-to-end pressure). Notice that it has no direction stipulation, and you'll get the same pressure against the walls if the fluid changes direction but flows with the same speed.

This particular question is asking about flow rate, which describes the fluid moving from one end of the pipe to the other end, so it calls for Poiseuille's Law, not Bernoulli's principle.

If they were to ask a question about "where is the pipe most likely to collapse?", then Bernoulli's principle would be the equation of choice, because that's an issue with the wall's pressure.