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
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~
