Fluids #2

A pipe carries water from a hot-water tank in the basement to the three floors above it.  Assume that both the cross-sectional area of the pipe and the water flow rate are constant.  Which of the following statements will be true about water exiting the pipe on the top floor of the building?

A.  The speed of the hot water will be higher than at the basement tank level.
B.  The speed of the hot water will be lower than at the basement tank level.
C.  The pressure of the hot water will be higher than at the basement tank level.
D.  The pressure of the hot water will be lower than at the basement tank level.

Answer: D

SHORT-AND-SWEET:

When you see a question that talks about different types of energy in a system, think of conservation of energy!  When applied to the flowing fluids, this principle is called the Bernoulli's principle.  The principle states that the sum of all the different kinds of energies of the flowing fluid will be constant.  What are the types of energy a flowing fluid can have?  Kinetic (because it is moving), potential (because it might be moving from one height level to another), and "pressure energy" (energy expended by fluid to exert pressure on the container walls).  Because their sum is always constant, an increase in one type of energy will come at the expense (decrease) of the other types of energy.

In the question above, water's potential energy will increase (because the water is flowing from the basement to the top floor) -- at the expense of either water's kinetic energy or pressure.  We are explicitly told that the flow rate (Q) and the cross sectional area (A) are constant, which means that the water velocity will remain constant as well (recall the expression for the flow rate: Q = A v).  That also means that there will be no change in kinetic energy.  Therefore, increase in the potential energy will have to come at the expense of the water pressure, leading to water pressure drop on the higher floors (answer D).

THE WHOLE STORY:

This week we continue to examine fluids.  More specifically, we are dealing with hydrodynamics: fluids in motion.  Like we said last time, when you hear the word "dynamics" think about kinetic energy.  But that's not all we have in this question.  Water moving from basement level to the top floor suggests some height change, which makes you think of gravitational potential energy.  It also talks about pressure.  Of course, you have to talk about pressure when you talk about fluids.

Though not explicitly stated, the question treats the building-water tank-pipe-water system as a closed system, in which there will be conservation of energy.  When we talk about conservation of energy for fluids, we are talking about Bernoulli's principle.

Bernoulli's principle applies to ideal fluids.  Let's recall characteristics of an ideal fluid:
- steady flow = all fluid molecules are moving at the same speed.
- incompressible = under compression, it will not change volume (due to repulsive forces between molecules).
- non-viscous flow = all molecules move linearly in laminar flow.

In the MCAT, unless explicitly told otherwise, assume that all fluids you encounter are ideal!

Back to good ol' Bernoulli and conservation of energy in flowing fluids.  What are the different kinds of energy that a flowing ideal fluid can have?  Well, for starters, given the "flowing" part, it can definitely have KINETIC ENERGY.  What else?  Think of a river flowing from a mountain spring towards an ocean.  It is flowing from a higher elevation to a lower ground, which points to it POTENTIAL ENERGY.  What else?  When we talk about fluids, we often talk about their pressure, which is essentially a measure of how hard the fluid molecules are pushing against the walls of the fluid container.  So, the fluid has to have some energy spared to exert pressure, so let's call it exactly that: "ENERGY to exert PRESSURE".

The conservation of energy principle says that in a closed system total energy will be constant.  Therefore, the sum of all the different kinds of energies will be constant as well.  For a flowing fluid that means:

ENERGY TO EXERT PRESSURE + KINETIC ENERGY + POTENTIAL ENERGY = CONSTANT

So, if the fluid flows faster and therefore expends more kinetic energy, it will have less energy to exert pressure on the walls of its container, and the pressure will drop (if the height does not change).

Let's use a more colorful example, that of a hamster (we'll name him Bernoulli) in a cage.
Bernoulli can have fun in his cage in several different ways.  He can run in the cage.  He can climb the walls of the cage (a pretty athletic hamster, huh?), or he can push against the walls of the cage (trying to escape, perhaps?).  Now, if Bernoulli wants to be running as fast as he can (kinetic energy), he won't have much energy left to be climbing the walls of his cage (gravitational potential energy).  If he decides to push, push, push against the walls of the cage, it will be challenging for him to be trying to run like a wind at the same time.

Now, let's look at the actual Bernoulli's equation:

The first element in the equation, P, is clearly the pressure of the fluid.  The second one looks familiar, too, doesn't it?  If you substituted the fluid density ρ for mass m, this part would look just like the kinetic energy expression that you had seen a million times.  The same thing with the last element in the equation, which looks much like the expression for potential energy (when you substitute ρ for m).

Now, let's go back to our question.  We know that in this system in which the water is moving from the basement tank to the top floor the total energy will be conserved.  So, whatever energy will be expended in the system to get the water to the top floor (potential energy) will come at the expense of either kinetic energy or the "pressure energy".  But which one will have to give?

Note that in the question itself you are told that neither the flow rate nor the cross-sectional area of the pipe change.  OK, so it's the same pipe, with the same cross-sectional area.  But what does the constant flow rate mean?  Flow rate tells us the amount of fluid flowing through a pipe in one second.  Constant flow rate means that whatever fluid volume enters the pipe in one second, the same volume of fluid will come out at the other end in that second.  The continuity equation also holds true in this case (1 and 2 refer to different points along the pipe, for example the basement and the top floor):

A1 v1 = A2 v2

If the cross-sectional area does not change (A1 = A2), it also means that velocity along the pipe will not change either (v1 = v2), which eliminates answers A and B.

However, it also means that fluid kinetic energy remains constant along the length of the pipe.  Therefore, the increase in the gravitational potential energy will come at the expense of the "pressure energy", and the water pressure on the top floor will be lower than in the basement tank (answer D).

BIG PICTURE:

1. When you see a question that talks about different kinds of energy, think conservation of energy (unless explicitly stated otherwise)!

2. Bernoulli's principle is the principle of conservation of energy applied to moving fluids.  Increase in one type of energy will come at the expense (decrease) of the other types of energy.  

~The MCAT POD Team~

Fluids #1

An open container filled with ideal fluid starts draining through a spigot, as shown in the image.

Which of these graphs best depicts how height H changes with the fluid velocity v at point 2?

ANSWER:  B.

SHORT-AND-SWEET:

This question requires as little as 20 seconds to answer.  Let's see how.  

What happens as the fluid starts draining?  The height H starts dropping.  What happens to the velocity v of fluid at point 2 as the fluid is draining?  It will decrease as well.  Therefore, H and v track together -- as one goes down, so does the other.  Look at the answer choices.  Answer choices A and C suggest the opposite, and answer choice D suggests that H is constant.  Only answer B suggests the correct relationship between H and v.  Because of conversion of potential energy (static conditions before the spigot opens) to kinetic energy (dynamic conditions once the spigot opens) the actual relationship will be: H = v^2 / 2g.

THE WHOLE STORY:

We start off with an ideal fluid inside an open container.  When the spigot opens the fluid begins draining at a certain velocity, and we can all agree that with that the height (H) of the fluid column starts decreasing.

The question asks about what happens to the height (H) of the fluid as the velocity of the fluid at the spigot (v) changes.

STOP!!!  What I will say next is one of the keys to success on the MCAT, and applies without exception, to every single MCAT section. 

After reading the question, take ten seconds to think about the problem intuitively.  

Chances are that by thinking about the question first, you will be able to guess what the answer should look like.  Then quickly scan through the answer choices, and eliminate those that make no sense.  

The MCAT (and medicine) is as much about picking the right answers, as it is about eliminating the wrong ones.  The reward for this strategy is more precious time left over for questions that actually require calculations.  In fact, most people who end up running out of time in the Physical Sciences section do so because they spend too much time "plugging and chugging" through questions unnecessarily. 

Let's go back to our question and see how we can apply this strategy!

We know that as the fluid is draining the fluid height H will be decreasing.  Initially the fluid velocity v will have some value.  But what happens to the fluid velocity as the container is actively emptying?  With less remaining fluid in the container, the fluid will be draining more slowly (decreased velocity).  

If this doesn't seem right, get a balloon, fill it with water, and punch a whole in the bottom.  Just don't tell your roommates that we told you to do this.

Back to our question.  We have concluded that as the fluid height decreases, the fluid velocity will decrease as well. 

By knowing this, and by looking at the answer choices, we can try eliminating some of the choices.  Choices A. and C. suggest an inverse relationship between fluid height and fluid velocity, which, as we established, is not the case.  Choice D. suggests that fluid height does not change with changes in velocity, which we also know is not the case.  

Therefore, the correct answer is B. 

..........Now, if the answer choices were such that we could not easily eliminate all of the wrong answers, here is the mathematical way to solve this problem. 

What you need to realize is that this is essentially a conservation of energy question.  How?  Before the spigot opens, the system (container with the fluid) is static, because there is no fluid motion.  Once the spigot opens, the fluid will start moving, transforming this into a dynamic system.  

STOP!!!  You should learn that whenever you encounter a question in which you have the same system initially at rest and then in motion, it is very likely that the underlying concept is conservation of energy, focusing on the exchange of potential energy and kinetic energy.  

Let's see how this principle applies here.  For any given fluid molecule its total energy will be the sum of its potential and kinetic energy.  If we take a molecule at the fluid surface (point 1, compared to point 2), given that the molecule is effectively not in motion, its total energy will be its potential energy:  Etotal = Epotential = mgH.

Once the fluid starts draining, by the principle of conservation of energy, each molecule's potential energy will transform into kinetic energy.  At point 2, the molecule will have only kinetic energy, so its total energy will be equal to its kinetic energy:  Etotal = Ekinetic = 1/2 mv^2.

Because total energy is conserved, potential energy at point 1 and kinetic energy at point 2 will be equal:  mgH = 1/2 mv^2

If we divide the whole equation by mass m, we are left with an equation which tells us about the relationship between the height H of the fluid column and the fluid velocity at the spigot v:  

H = v^2 / 2g

This means that the height H of the fluid column is proportional to the square of v.  

You should be able to recognize that what we have in this question is a (parabolic) square function, which is depicted in answer B.

BIG PICTURE:  

1.  Think first, calculate second!  Or, more accurately, think (and eliminate) first, calculate (if needed) second.

2.  Recognize when a question asks you to recognize static ("resting") and dynamic ("moving") conditions within a system.  When they are both in the question, think POTENTIAL and KINETIC ENERGY, and CONSERVATION OF ENERGY!

3.  Know how to interpret graphs, and we mean KNOW IT!