four modeling instruction fluids modules

U16-Multiparticle Model: Buoyancy and Pressure

Unit 16: Fluid “Statics” By the time you finish all lab and class activities, you should be able to: 1) A fluid is a state of matter where the individual atoms/molecules do not have fixed positions

relative to each other. 2) The mass density, ρ, of matter is described by the amount of mass, m (kg), that is contained with a

given volume, V (m3): Density [kg/m3] = mass [kg]/Volume [m3]: ρ = m/V

3) The pressure at a particular point in a fluid is the energy per unit volume (energy density) that must be transferred from another system into the fluid system in order to create a unit volume of fluid at that point. An alternative definition is the force per unit area exerted by an external force on the volume. Pressure is a scalar property of fluids:

€

pressure[Pa] =Energy[J ]

Volume[m3]=

Force[N ]Area[m2]

4) 1Pa = 1pascal = 1N/m2. Other units for pressure and their relation to one atmosphere: 1 atm = 14.7 lbs/in2 = 1.013 bar = 29.92 in Hg = 760 mm Hg = 34 ft H2O

a) Absolute Pressure: Pressure measured from zero = vacuum. b) Gauge Pressure: Pressure measured from zero = local atmospheric pressure. c) Atmospheric pressure at sea level is 101,300 Pa ≅ 105 Pa.

5) Archimedes Principle states that the buoyant force (force pushing opposite the direction of the gravitational field) acting on an object immersed in a fluid is equal to the weight of the displaced fluid.

Fbuoyant = Fg-displaced fluid = mdisplaced fluidg = ρdisplaced fluidVdisplaced fluidg

Theory – Fluid Statics Examination of fluids is conveniently broken down into the use of the four models we discussed last semester, namely the free particle, constant force particle, impulsive force particle, and central force particle. The slight problem that we have to deal with (and this is a gross understatement) is that we are now dealing with a multi-particle system. At most we dealt with two particles, for example in collisions. In physics it is well known that are no “closed” solutions to any problems involving three or more particles, i.e. these problems can only be solved by numerical approximations. So why do we bother? There are lots of practical reasons, for example to help explain the behavior of the air that we breathe and the water essential for maintaining life on this planet. Fortunately we can approximate the behavior of fluids with macroscopic quantities, such as pressure, density, viscosity, etc. Our first lab activity is to characterize a relationship between pressure difference and depth or density for a variety of common liquids. How do we characterize a given liquid? Here is a list of things you might want to consider:

• Type of liquid • How deep we take measurements • Volume of the liquid • Stickiness of the liquid

Pressure

A V ρ

Δy


In the lab we will focus on a few easily controlled quantities, two of which are shown below, the depth “Δy” (solid line) at which we measure the pressure difference ΔP, and the density “ρ” (dashed line) of the liquid involved. Typical data is given for the lab situation at right. The solid line was measured in water, the dashed line was measured at a constant depth. Since this is another multivariable lab, like the uniform circular motion lab, we will have to keep in mind what the constants of the system are. E.g. for the data at right ΔP vs. Δz, the density of the liquid water, ρ=1000kg/m3, is constant. Based on the data above, if we analyze the SI units of the slope we find the for pressure versus depth (Pa/m)=(N/m3)=(N/kg)(kg/m3), or units of gravitational field times density. For pressure versus density (Pa/kg/m3) = (N/kg)(m), or units of gravitational field times depth. Furthermore the slopes are related to the product of those two constant. The consensus result of this analysis is the “hydrostatic pressure” equation: ΔP = ρgΔz Derivation: This relationship can be derived from first principles by examining the system at right. Imagine placing a box of area “A” and height “Δz” in a pool of fluid (the box inside the pool is also filled with the fluid). The pressure of the fluid on the top of the box is Fgt/A and the pressure of the fluid on the bottom of the box is Fgb/A. The weight of the fluid/area inside the box is mg/A. Since the box is not moving, and behaves like a free particle, these forces must balance:

Fgb/A = Fgt/A + mg/A Recall F/A = P and m = ρV

Pb = Pt + ρVg/A Recall V = AΔy

Pb = Pt + ρAΔyg/A = Pt + ρgΔy Lastly since ΔP = Pb-Pt

ΔP = ρgΔy This result can be extended to a discovery made by Archimedes that the buoyant force on an object (partially or completely) submerged in a fluid is equal to the weight of “displaced” fluid. The force is simply the pressure difference times the area of the box.

ΔPA = ρgΔyA But since ΔPA = F and ΔyA = V

Fbuoyant = ρgV = mg = Fg, displaced water

The hydrostatic pressure relationship can also be extended to “Pascal’s Principle”: Pressure applied to an enclosed fluid is transmitted undiminished to every point of the fluid and to the walls of the containing vessel. Therefore, if we examine the pressure in a “U” tube at the same height (Δz=0) where one side of the tube has a large cross sectional area than the other (see diagram at right), and since ΔP=P2-P1 = 0, the force/area at either side of the tube must be the same:

F1/A1 = F2/A2

A

Fgt

Fgb

Fg

F1

A1

F2

A2

P(Pa)

103000

0 1500 ρ(kg/m3) Δy(m) 0.5


Example 1) Consider an 80 kg person with a density of about 800 kg/m3 submerged in water.

a) Provide an interaction and force diagram of the person in the water:

b) Determine the volume of the water displaced by the person: V = m/ρ = 80 kg/(800 kg/m3) = 0.1 m3 (100 liters).

c) Determine the force of the water pushing the person up: F = mg = ρwaterVg = (1000 kg/m3)(0.1 m3)(10 N/kg) = 1000 N.

d) Would this person float or sink in the water if they were released? Answer: Float since the density of the person is less than that of water.

e) If the person dives down 10 m, what will be the pressure on their eardrums? Answer: Our hydrostatic pressure relationship ΔP = ρgΔz can be re-written:

Peardrums = Patm + ρgΔy = 101300Pa + (1000kg/m3)(10N/kg)(10m) = 201300Pa The pressure nearly doubles. 1300 Pa is insignificant to atmospheric pressure?

f) What is the pressure difference inside the person’s eardrums if they haven’t tried to “equalize” the pressure in their head with the water outside?

ΔP = ρgΔy =(1000kg/m3)(10N/kg)(10m) = 100000Pa g) What is the force on the person’s eardrums (assume to be a circle of 1cm radius)

Recall the area of a circle = πr2 and that pressure P = F/A P = F/A = F/πr2

Rearranging: F = Pπr2 = (100000N/m2)π(0.005m)2 = 7.8N

This does not seem like much, but it equivalent to turning your head sideways and resting about 8 hamburgers on a 1cm diameter wooden dowel on your eardrum. OUCH!

h) If the person is allowed to float, what volume of the person would be above the surface of the water?

What information do we have? In solving a problem it is always good to start by writing down what you know:

mass of person: 80kg density of person (ρp): 800 kg/m3 volume of person (Vp solved): 0.1m3 weight of person (solved): 800N density of water (ρw): 1000 kg/m3

The person’s volume includes that which is above and below the water surface: Vp = Vpa + Vpb

The volume of the person below the water equals the displaced water volume: Vp = Vpa + Vw

person

water (or air) earth Fg

Fb,w

Fg

Fb,a

FS


Rearranging: Vpa = Vp – Vw

The forces must still balance: Fg,w = Fg,p

ρwVwg = ρpVpg Rearranging:

Vw = ρpVp/ρw = (800kg/m3)(0.1m3)/(1000kg/m3) = 0.08 m3

Therefore: Vpa = Vp – Vw = 0.1m3 – 0.08m3 = 0.02m3

i) Determine the force air on the same person standing on dry land. Less obvious is that air also exerts a buoyant force on us each day. But since air is about 1000 times less dense than water, this buoyant force equals

F = mg = ρairVg = (1.4 kg/m3)(0.1 m3)(10 N/kg) = 1.4 N. Note that in this case the buoyancy force is a tiny fraction of the force balancing the person’s weight (in this case 800N), that is assumed to be balanced by some kind of surface force. The buoyant force of air is approximately equal to the weight of one hamburger. Still, air is a good thing.


Operational Definitions: Fluid: Density (units): Buoyancy: Pressure (units): Hydrostatic Pressure: Pascal's Principle


Conceptual Worksheet C: Buoyancy 1) A metal sphere is found floating in a pool. Is this logical? What is plausible explanation for this

observation?

a) Draw an interaction and force diagram with an explanation of the forces on the sphere.

b) The sphere is 1m in diameter. If it is half submerged determine the weight of the sphere. Use your force diagram to help you determine its weight. You may assume g=10N/kg and ρwater =

1000kg/m3.

€

Vsphere =43πr3 where the diameter = 2r. Show all work.

2) A float is tied to the end of a rope, both of which are submerged in

seawater as shown in the picture at right. a) Draw an interaction diagram and a force diagram on the float.

Label all forces on both diagrams.


b) The mass of the float is 50.0-kg and it has a volume of 1.0m3 and the density of the seawater is about 1000kg/m3. Determine the density of the float and the tension on the rope. You may assume g = 10.N/kg for this activity. Please show all work.

c) Plot buoyancy versus fluid density if the float were immersed in the following fluids (vegetable oil, water, corn syrup, and liquid mercury.) Use the internet to determine the fluid densities and generate a mathematical model explaining what the slope means.

3) A spherical helium-filled balloon is attached to a string that

is fixed to the ground (see diagram). The balloon has a volume of (0.30/4) m3 and helium has a density of ρ = 0.20 kg/m3. The balloon is floating in air (ρ =4/3 kg/m3.) a) Draw a multi-particle model diagram to describe the

pressure inside and outside of the balloon and then explain your model in words below.

15000

Fb(N)

0 ρ(kg/m3)


b) Interaction diagram the system and force diagram the balloon clearly labeling your interactions and forces.

c) Provide a math model to solve for forces on the balloon if up is considered the positive direction, clearly identifying names of the forces on the balloon. The solution should use ± instead of vector notation.

d) Is this a free particle or constant force particle? EXPLAIN YOUR ANSWER.

e) Determine the buoyant force Fb from Archimedes Principle.

f) What is the weight of balloon and helium if the tension on string is 0.8N?

g) What is gravitational force on the helium alone?


Basic Worksheet B: Buoyancy 1) Read the description at right and then explain

if the new reading on the scale is more, same, or less than 100N.

2) The experiment is repeated with the six

different blocks listed below. In each case, the blocks are held completely submerged in the water.

a) Rank these blocks on the basis of the

scale reading when the blocks are completely submerged, from largest to smallest. Support with force diagrams.

b) If the blocks were released while submerged, which, if any, would sink to the bottom of the flask? Provide schematic force diagrams for one floater, one that IS sinking, and for one already on the bottom.


3) A basketball has a mass of 0.50kg and a volume of 8x10-3m3. What is the magnitude of the force on a basketball needed to keep it fully submerged in water? Start with an interaction diagram and then force diagram, draw the force diagram to scale if you can.

4) A 10cm x 10cm x 10cm wood block with a density of 700kg/m3 floats in water. What is the

distance from the top of the block to the water if the water is fresh? Always interaction and force diagram first.

5) A rectangular steel box just barely floats in glycerol (look up glyercol's density). The density of the

steel is 7600kg/m3. The bottom plate of the steel is 1cm thick x 10cm by 20cm. The height of the sides are 9cm. What is the thickness of the steel wall on the sides?


Useful constants: gravitational field g=10N/kg, density of water, ρwater = 1x103 kg/m3. The three problems below involve the picture at right. A boy sits inside a plastic ball, submerging a portion of the plastic ball in the water. This is a form of entertainment in China. 6) What is the correct (though NOT to scale) force diagram of the ball? Remember that the boy is

acting ON the ball and NOT part of the ball! 7) If the weight of the water (V=0.050m3) pushing up the boy equals the weight of the boy, calculate

the boy's mass. a) m = 40kg b) m = 25kg c) m = 50kg d) m = 20kg

8) Which of the following diagrams most accurately depicts the density of fluid molecules in and

outside the ball (ignoring the boy). 9) 1kg cylinders of aluminum (ρAl = 2700kg/m3), copper (ρCu = 8960kg/m3) and lead (ρPb =

11350kg/m3) are dropped into ethanol. Rank the buoyant force of the ethanol on these objects. a) Cannot be determined from the information provided. b) Fb,Al > Fb,Cu > Fb,Pb c) Fb,Pb = Fb,Cu = Fb,Al d) Fb,Pb > Fb,Cu > Fb,Al

Fg,boy Fg,ball

Fb,water Fb,air

Fg,boy Fg,ball

Fb,water

Fg,boy

Fb,water Fb,water

Fg,ball

Fb,air

a) b) c) d)

a) b) c) d)


The following three problems deal with this description: A clean cylindrical glass of height "H" is turned upside down and pushed down into an oil well (ρ=900kg/m3). The glass is carefully lifted out and the level "h" of air is measured (diagram at right). The temperature of the oil is the same throughout the well. Up is positive. 10) The depth at which the oil reaches height "h" inside the glass is defined as z=0.

Which of following graphs correctly describes pressure as a function of depth according to this reference?

11) If the volume of air (treat as an ideal gas) in the glass at depth is one third that at the surface of the

oil well, to what depth was the glass pushed? Let Patm=1x105Pa. a) (200/9)m b) (300/9)m c) (400/9)m d) (500/9)m

12) What is the algebraic relationship between "h" and the height of the glass "H" at depth "Δz" in oil

of density "ρ" in the earth's gravitational field "g"? 13) Earth's atmospheric pressure does not decrease linearly with increasing altitude because air is

compressible. Find the mathematical model that best fits the data at right. Find the decay length of the earth's density (37% of max).

a) b) c) d)

+z 0

P(Pa)

+z 0

P(Pa)

-z 0

P(Pa)

+z 0

P(Pa)

a) b) c) d)

h H

€

h =H

ρgΔy

€

h =PatmH

Patm + ρgΔy

€

h =(Patm + ρgΔy)H

ρgΔy

€

h =ρgH

Patm + ρgΔy


Advanced Worksheet A: Hydrostatic Pressure 1) Identical spaceships are sent to eight planets. After landing

on each planet pressure readings are taken at the same fixed depth in a column of water inside the spaceship. The data is presented at right. Please show all work. a) What is the math model for this data? The largest

pressure is 226,000 Pa. Clearly identify your slope and intercept.

b) What is the pressure in any of the spaceships when it is far from any planet? Explain how you got this answer from the graph above.

c) At what depth was the pressure sensor placed in the column of water (ρ=1000kg/m3)? Show all work.

d) Draw a multi-particle force diagram describing the pressure difference due to a large gravitational field and a small gravitational field. Explain your answer.

High g Low g


2) A research submarine has a 10-cm-diameter window that is 8.4 cm thick. The manufacturer says the window can withstand forces up to 1.2×106 N. What is the submarine's maximum safe depth in salt water? The pressure inside the submarine is maintained at 1 atm. Support your answer with a multiple particle diagram. Treat the water molecules as tiny hard spheres.

3) The container shown in the figure is filled with oil

(ρ=900kg/m3). It is open to the atmosphere on the left hand side of the tube. a) What is the gauge pressure at point A? Express your

answer using two significant figures.

b) What is the pressure difference between points A and B?

c) What is the pressure difference between points A and C?

d) Support your answer with a multiple particle diagram.

4) A friend asks you how much pressure is in your car tires. You know that the tire manufacturer recommends 30 psi, but it's been a while since you've checked. You can't find a tire gauge in the car, but you do find the owner's manual and a ruler. Fortunately, you've just finished taking physics, so you tell your friend, "I don't know, but I can figure it out." From the owner's manual you find that the car's mass is 1500 kg. It seems reasonable to assume that each tire supports one-fourth of the weight. With the ruler you find that the tires are 15 cm wide and the flattened segment of the tire in contact with the road is 13 cm long.


Unit16: Buoyancy Deployments Group Members: You are given a block of wood and access to all the usual experimental tools of the lab. Find the buoyant force on the block when it is allowed to float on the surface of the water. In the space below show ALL your representations: Data (no graph required – only one data point), diagrammatic, mathematical, verbal (written description of how you solved the problem) and experimentally determined buoyant force. AFTER you have written up your group answer and handed it in, whiteboard your results for class discussion. A plastic fish attached to a syringe, a model for a fish swim bladder, can be made to sink or rise by squeezing or releasing the bottle in which the "fish" swims. Objectives: To provide diagrammatic, graphical, written and mathematical solutions for the volume of the plastic fish. 1) Force Diagram the fish at the three instants shown. A) Floating freely at the top of the water, B) sinking faster and faster, and C) at rest on the bottom of the container. A) Sinking faster and faster starting at the top of the water, B) sitting still, and C) at rest on the bottom of the container. A) Pushed down by the plastic container B) sitting still, and C) rising faster and faster from the bottom of the container. Below identify the labels of all forces and the magnitude and direction of the sum of forces at each point. ΣFA = ΣFB = ΣFC = Reminder: The weight of the fish MUST be constant on your force diagrams. 2) In clear, concise sentences write one paragraph explaining how intend to solve for the volume of the plastic fish.

A

B

C


3) At "neutral buoyancy" ΣF=0 when the object is completely immersed in the water and not touch anything else except water. Set up a mathematical equation for neutral buoyancy that will solve for the buoyancy of the fish and then its volume. 4) Use your result above and appropriate measurements to determine the volume of the plastic fish ONLY (no air in the syringe).

Weight of fish in air: ________ Volume of air bubble:_______ Predicted fish volume: _______ Actual fish volume: _______

U17-Multiparticle Model: Ideal Gas Law and Kinetic Theory

Unit 17: Ideal Gas By the time you finish all lab and class activities, you should be able to understand that: 6) A gas is a fundamental state of matter where the individual atoms or molecules do not have fixed

positions relative to each other and typical are far apart from each other. 7) The ratio of number of particles, N (typically a freakin' huge number) to the volume in which the

gas is contained, V (m3), is defined as the particle density,

€

N :

€

ParticlesVolume[m3]

≡ Particle Density[m−3]→NV≡ N

Particle number is referenced to Avagadro's number NA = 6.022x1023 pls. If you have this number of avacados it is called one guaca-mole.

8) The ratio of a force exerted over an area, or the work done F∆x to compress the volume A∆x is defined as Pressure. Pressure is a scalar property:

€

Force[N ]Area[m2]

=Energy[J ]

Volume[m3]≡ Pressure[Pa]→

FA

=EV≡ P

Consequently pressure is an example of an energy density, reflecting the internal energy of the gas due to the kinetic energy of all the atoms or molecules in the gas. Internal energy of a gas is the result of translational, rotational, and vibrational motion of the particles. The latter two forms of motion can only happen for molecules (minimum of two atoms).

9) In a confined volume, the finite size of the particles means an enormous number of ELASTIC collisions take place between gas particles and the container or between gas particles at any finite interval of time. Gas pressure reflects conservation of linear momentum.

10) INELASTIC collisions involve special kinds of physics or (God forbid) chemistry. 11) Internal energy is proportional to the temperature "T" of the gas, the number of particles "N" and a

proportionality constant called the Boltzmann constant "kB" = 1.381x10-23 J/K, reflecting the statistical description of average particle collision behavior. Units of temperature: a) Absolute scale = Kelvin (K). 0K is the classical ceasing of all particle motion, and the triple

point of water is 273.15K. Note that no degree sign (°) is used with Kelvin. b) Centigrade scale = Celsius (°C). Water freezes at 0°C and water boils at 100°C. The range

between the two is 100C° (note the order, a temperature is °C, a temperature difference is C°). c) Fahrenheit scale = Fahrenheit (°F). Water freezes at 32°F and water boils at 212°F. The story

behind this scale is interesting, based on the freezing point of "brine" and body temperature. 12) The ideal gas law in physics relates the work done by/on an expanding/compressing gas "PV" to the

internal energy in "N" particles at a temperature "T" and proportionality constant "kB": PV = NkBT

13) 1Pa = 1pascal = 1N/m2. Other units for pressure and their relation to one atmosphere: a) 1 atm = 14.7 lbs/in2 = 1.013 bar = 29.92 in Hg = 760 mm Hg = 34 ft H2O b) Absolute Pressure: Pressure measured from zero = vacuum. c) Gauge Pressure: Pressure measured from zero = local atmospheric pressure. d) Atmospheric pressure at sea level is 101,300 Pa ≅ 105 Pa.

Theory – Ideal Gas

The particle theory of matter is one of humankinds greatest achievements. Why? Because empirical evidence, such as the kind you are about to collect in lab, was used to infer the presence of things humans could not directly see, namely atoms undergoing random collisions. In the 1850s French physicist Émile Clapeyron combined results from English physicist Robert Boyle and French baloonist


Jacques Charles into what we now call the Ideal Gas Law (PV=nRT)1. n=N/NA is the number of moles of "N" particles and "R" was the experimentally determined "ideal gas constant", 8.314 J/(moleK). Later this emperical result would be firmly rooted in "kinetic theory" developed by Austrian Physicist Ludwig Boltzmann2 (Figure 1) who defined the ratio of the ideal gas constant to Avagadros number as a fundamental statistical mechanical parameter that would later bear his name, the Boltzman Constant:

€

RN A

= kB =1.381x10−23 JK

The importance of this result is that humans had figured out how to measure pressure and temperature, the result of gazillions of colliding particles, that could not be seen! Physicists are more partial to the form of the ideal gas law PV=NkBT because it emphasis the collective result of gazillions of particles elastically colliding with each other and with the walls of container of volume "V". "NA" is an inconceivably huge number even for objects as small as a grain of sand. One NA of beach sand will fill a sandbox approximately 1km wide, 1km long and 1km deep. Reasonable volumes and masses of a mole of particles are obtained only when dealing with objects as small as atoms.

Clapeyron Boyle Charles Boltzmann

Figure 1 That's enough of history. What does kinetic theory mean and why should you care? It turns out that understanding fluids requires a grasp of the microscopic picture. If you can understand why gazillions of interacting particles give rise to pressure, then you are more likely to understand more challenging aspects of fluids, such as hydrostatic pressure, dynamic pressure due to moving particles, and real physiological problems such as high blood pressure. One of the important leaps of faith we make when dealing with gasses is that temperature is a measure of kinetic energy corresponding to approximately the average speed of a gas particle of mass "m". The result of kinetic theory is that:

€

m < v2 >3

= kBT

<v2> is the "root mean square" speed of the particles in the container, a way of describing an average speed of all particles in the volume. The 1/3 comes about because on average 1/3 of the particles are going in any given coordinate direction, x, y or z. The term to the left then reflects a portion of the average kinetic energy of a gas particle, which is proportional to the temperature of all the particles in the container – a truly impressive result. Only within the past 10 years have humans been able to control the temperature of an individual atom3, so kinetic theory was amazing, exceeding humans abilities to control individual atoms by over 150 years! Because the ideal gas law has four state variables (P, V, N, T) we will need to describe processes in which some of the variables are kept constant. For example, constant pressure processes are called "isobaric". Constant volume processes are called "isochoric" or "isovolumetric". A constant number of particles will be a "constant number of particles" (duh) or isomolar particle number. Lastly a constant temperature is called an "isothermal" process. To help you visualize the features of the ideal gas we will introduce a set of powerful diagrammatic tools (Figure 2) that will attempt to make sense of 1 http://en.wikipedia.org/wiki/Ideal_gas_law 2 http://en.wikipedia.org/wiki/Kinetic_theory 3 http://physics.aps.org/articles/v5/121


inconceivably huge numbers of particle interactions. The table below summarizes how to describe each term in the ideal gas relationship. Particle number and volume are pretty simple to diagram. Temperature and pressure require a bit more thought, especially when those variables are changing.

Term SI units Diagram Particle # "N"

Later on we will find it advantageous for quick drawing purposes to simply associate one arrow with one particle, without a "dot".

Pure #

Volume "V"

Please note that the volume could be a square or any other easy shape to draw. The size difference is the key feature.

Cubic meters "m3"

N.B.: The "cgs" unit is

the cubic centimeter

which can be written as:

cc = cm3 = mL

Temperature "T" For drawing

purposes we will dispense with color coding and simply infer longer arrows

implies more energy and thus

higher temperature.

Kelvin "K" N.B.: This is

the only correct unit of temperature to use with the

ideal gas law.

Pressure "P"

Key:

A "v" represents a collision with the

wall. An "X" represents two "v"s,

i.e. two particles colliding.

Pascal "Pa" N.B.: The number of

particles is 14 in both low

and high pressure – can you see why?

Piscatorial "F" Considered to be a

hidden fishy variable in the ideal

gas law.

Seussian "S"

Figure 2

Small volume: Large Volume:

Low temperature: High temperature:

Low pressure: High pressure:

Few particles: Many particles:

=X =v


Pre-lab demonstration activities: The instructors will have you examine two introductory activities to get you thinking about how ideal gasses behave. The first will be "Handboilers", sealed glass tubes filled with small amounts of dyed alcohol. These are delicate instruments so please be very careful as they break easily. As you hold the bottom or top of each boiler what do you observe? What do you think is going on? You will be asked to explain the behavior in terms of the diagrammatic tools shown on the previous page. The next activity will be to carefully observe the opening of a fresh bottle of seltzer. The instructor may use a camera to help magnify the activity. What do you observe? Why? Again, as part of the pre-lab discussion use the diagrammatic tools to explain what is going on. Our first lab activity is to characterize a relationship between the various ideal gas law terms: Pressure, Volume, Particle number and Temperature. As an example we will present one result that will NOT be examined in lab, in part because it is hard to do. But we can describe the experiment in easy to understand terms and even model some parts of the relationship with a PHET simulation on ideal gasses (http://phet.colorado.edu/en/simulation/gas-properties). The experiment involves a glass bottle fitted with a stopper that has a small hole in it to let gas in and out, so the volume is constant. The bottle can be immersed in a bath of liquid that can be cooled to the boiling point of nitrogen (77K), cooled to the freezing point of water (273K) and then heated to the boiling point of water (373K). Because there is a small hole in the bottle, dry gas (free of moisture) can move in and out of the bottle as necessary to keep the atmospheric pressure constant in the bottle. The remaining parameters that can be changed are clearly the number of particles (dependent) and temperature (independent). We will first diagram what we expect of the gasses at low and high temperature, and then plot graphically how particle number depends on temperature from a virtual experiment. Notice the diagram on the left side involving two same size circles (constant volumes). As the temperature increases the number of the interactions with the chamber and other particles (i.e. the pressure) stays constant by pushing out particles through the hole, such that the number of lines (particles) decreases in the bottle. The plot at right, from ideal data, is an inverse. This experiment is difficult to accomplish because measuring such a large number of dry gas particles must be inferred and cannot be measured directly, plus it is essential to have a wide range of temperatures so that you can see the inverse nature of the plot. Linearization of this data would yield the following math model:

€

N (# pls) =7.3x1025(# pls * K )

T ( K )=

PVkBT

Check this result from the known constants to see if the slope is correct. If we could measure the number of particles and temperature accurately, we could directly extract the Boltzmann's Constant from the slope of the linear plot and the known constants of pressure and volume.

T(K) Constants: Volume = 1L = 0.01m3 Atmospheric Pressure = 101300Pa

0 500

N(#)

7.4x1023


Constant Volume and Pressure


Conceptual Worksheet C: Ideal Gasses

4) The volume (squares below) and number of gas particles (16) in a container are kept constant.

Diagram the situation at low and high temperature and sketch what a graph of pressure versus temperature should look like, labeling all axes appropriately and explaining what the slope would equal.

5) The volume (squares below) and temperature in a container are kept constant. Diagram the

situation at small and large particle number and sketch what a graph of pressure versus particle number should look like, labeling all axes appropriately and explaining what the slope would equal.

6) The pressure and number of gas particles (16) in a container are kept constant. You choose which

parameter to control (temperature or volume). Diagram your and graph your results, labeling all axes appropriately and explaining what the slope would equal.


Constant Volume and Particle Number

Small N: Large N:

Constant Volume and Temperature

Constant Pressure and Particle Number


7) The temperature and number of gas particles (16) in a container are kept constant. Diagram the

pressure at low and high volumes and graph what pressure versus volume should look like, labeling all axes appropriately and explaining what the slope of the linearized data would equal.

8) Diagram how small and large particle numbers would demand of the volume such that the pressure

and temperature in a container are kept constant. Sketch what a graph of V versus N should look like, labeling all axes appropriately and explaining what the slope would equal.

9) Between the example and the five problems above, which of the P,V,N,T comparison's have we left

out? Decide which terms are constant and diagram and graph the variables and explaining what the slope would equal.

Small Volume: Large Volume:

Constant Temp and Particle Number

Small N: Large N:

Constant Pressure and Temperature

Constants:


Basic Worksheet B: Ideal Gas, Buoyancy, and Hydrostatic Pressure 5) Recall the opening of the fresh bottle of soda water from the beginning of this unit.

a) First draw a diagram and graph of the parameters being kept constant and varied in this demonstration:

b) Calculate the number of gas molecules inside a 3mm diameter bubble using reasonable estimates for the remaining constant state variables (P and T).

c) What is the internal energy of the gas molecules inside the bubble? Compare it to the work required to lift a 100g apple 1m up.

6) This is a very subtle and tricky problem that you now have enough

information to solve. Consider an air-filled balloon weighted so that is on the verge of sinking, that is, its overall density just equals that of water. a) If you push it beneath the surface it will do what and why?

b) Sketch the molecular interactions and size of the balloon at the surface and at depth in the space below. Explain why your diagram makes sense at depth?

c) At what depth would the volume of the balloon be half of its volume at the surface of the water? What must you assume to solve this problem? You may assume a 1L balloon and STP at surface, but can be solved without this information.

Constants:


7) Calculate the volume of Avagadro's number of gas molecules at STP.

a) How big is this volume? Compare it to ordinary household items.

b) How much internal energy is stored in this much ideal gas? Compare it to the work required to lift a 100g apple 1m up.

8) From the following data complete the following questions:

a) Graph and linearize the data at right as needed.

b) Write down the mathematical model explaining the slope and intercept in terms of ideal gasses:

c) How much internal energy does the system have?

d) If this corresponds to a mole of gas particles, calculate the temperature they are at.

V (m3) P (Pa) .1 40 .5 8 1 4 2 2 4 1 5 .8 8 .5 10 .4


9) Below is a diagram of air molecules at a density of 1.3kg/m3 inside the crew volume (1.0x103m3) of

a submarine when the submarine is at the ocean's surface. Note only two of the molecules colliding with the walls of the ship – that will be a model for atmospheric pressure in the sub.

a) Determine the mass of air in the submarine. How many significant digits here?

b) Determine the number of air molecules (MW = 29g/mole).

c) On August 12, 2000, the Oscar II class Russian submarine Kursk sank in 100m of water near the Finish/Russian border, when the fuel tank of one of its own torpedoes exploded. All hands were lost. What was the pressure of the air in the submarine assuming all the air was trapped by the exposure to seawater at 100m depth?

d) Use the ideal gas relationship to draw the volume the air trapped in the sub at 100m depth, employing the same number of air molecules as in part "a" to explain diagrammatically what happens to pressure in the sub.

e) Explain if air is a good example of an ideal fluid in this situation.


Advance Worksheet A: Ideal Gas, Buoyancy, and Hydrostatic Pressure

1) Recall the "handboiler" from the beginning of this unit.

a) First draw a diagram and graph of the parameters being kept constant and varied in this demonstration:

b) Using a reasonable guess for the volume of gas inside the handboiler and assuming the number of gas particles in the boiler when it is at room temperature correspond to atmospheric pressure, estimate the pressure of the gas inside the toy when at thermal equilibrium with your hand.

c) What is the internal energy of the gas at that temperature? Compare it to the work required to lift a 100g apple 1m up.

2) A typical hot air balloon can be modeled as a sphere with a radius of about 8.8m. The temperature

of the air inside the balloon is heated to around 210°F. a) Determine the number of ideal gas particles inside the balloon. What assumption did you have

to make to solve the problem?

b) Determine the lift capacity (weight of balloon and passengers) of the balloon at STP.

Constants:


3) Earth's atmospheric pressure does not decrease linearly

with increasing altitude because air is compressible. The temperature changes dramatically with elevation too. a) How do high altitude balloons account for this

difference? Diagram below the "volume" of a high altitude balloon released at the surface of the earth and again when it reaches and altitude of 15km.

b) A weather balloon filled with helium (ρ = 0.164kg/m3) and can lift a payload of 100kg. Calculate the radius of the spherical balloon at the surface of the earth assuming about one atmosphere of pressure and the temperature from the chart above.

c) Calculate the number of helium particles in the balloon.

d) Calculate the volume of the envelope at an altitude of 15km.

e) Calculate the pressure inside the balloon at 15km.


4) A diver 50 m deep in 5°C fresh water exhales a 0.50-cm-diameter bubble. The following steps will

help you determine the bubble's diameter just as it reaches the surface of the lake, where the water temperature is 20°C. Assume that the air bubble is always in thermal equilibrium with the surrounding water. a) Determine the pressure of the air inside the bubble at depth.

b) Determine the volume (and therefore diameter) of the bubble at the surface of the lake using the ideal gas law and pressure at the surface of the lake.

c) Beer bubbles and soda bubbles double or triple in diameter as they float upwards a mere 20cm in a beer stein or soda bottle. Explain if hydrostatic pressure can be used to explain the size change.

U18-Multiparticle Model: Ideal Fluid Flow

Unit 18: Fluid “Dynamics” By the time you finish all lab and class activities, you should be able to: 1) Written in terms of variables that describe a fluid, the statement that all changes in energy must sum

to zero is called Bernoulli's equation:

€

0 = ρgΔy +12ρΔ v2( ) + ΔP

a) For this expression of conservation of energy, the fluid must flow in smooth streamlines without friction, must not have any vortices, and must be incompressible. These conditions describe an “ideal fluid” but are fairly good approximations for many fluids.

2) For steady state continuous flow confined in a pipe or tube, the volume flow rate “ΔV/Δt” must remain constant in time and be the same at any cross section. If the cross-sectional area “A” changes, then the speed of the fluid “v” must change to keep the volume constant. For incompressible fluids, the volume flow rate is given by:

ΔV/Δt = A1v1 = A2v2

a) and is an expression of another conservation law--the conservation of fluid volume, which can neither appear nor disappear in a closed system.

Theory – Fluid Dynamics

As mentioned at the beginning of the last unit: “Examination of fluids is conveniently broken down into the use of the four models we discussed last semester, namely the free particle, constant force particle, impulsive force particle, and central force particle.” The complication is that these models are to be applied to multi-particle problems in which macroscopic phenomena (like pressure, density, etc.) are used to describe the collective response of all the particles. In the previous unit we examined problems in which the collective motion of the particles was zero, that is the average particle was interacting with its neighbors, but not in motion with respect to the laboratory reference frame. In this unit we examine free particle behavior for particles in motion at constant velocity with respect to the laboratory reference frame. Though this velocity may be different at different points in the system, for the time being we will ignore acceleration. Our first lab activity examines the influence on pressure due to the motion of particles. When a tube is attached to a sensitive pressure gauge and the tube is swung in circles at constant speed, the pressure inside the tube can be observed to decrease (diagram at right). If we examine the pressure difference versus speed we find the following results in the graph below. The graph appears to be a downward opening parabola that can be linearized by plotting absolute pressure versus velocity squared. As with all previous models we examine the slope and intercept of the linearized data to understand the physics of the problem. Clearly the pressure depends on the velocity squared. The same principle is used by airplanes ("Pitot" tube) to determine air speed.

The units of the slope are (Pa/m2/s2) = (kg/m3). The slope must be related to any constant(s) of the experiment. The pressure the sensor is measuring is due to loss of air molecules inside the tube as it swings around. This loss can be visualized a consequence of a lower pressure region at the end of the tube due to turbulence as the tube sweeps away air molecules from its path. The higher pressure air inside the tube will push air molecules into the lower pressure region to maintain equilibrium.

- +


The intercept also has important physical consequences and can be easily related to the pressure in the room when the tube is not in motion. The density of air is between 1.2 and 1.4 kg/m3, depending on temperature and pressure conditions. The slope will have about the same order magnitude as the values quoted, but if the data is carefully collected, should be off by a factor of two. Just like the energy versus velocity lab resulted in a linearized data plot with slope of half the mass of the cart, this linearized data will result with a slope of half the mass/volume of fluid. Ultimately we develop a consensus mathematical model:

P = Patm-(1/2)ρv2

where the reference pressure, “Patm”, is atmospheric pressure in the room and the speed of the air, “v”, is at the end of the tube and “ρ” is the density of the air. The hydro”statics” and hydro”dynamics” labs can be combined into a single relationship developed by Bernoulli for the situation in which the reference fluid is in motion and changing height:

ΔP = -ρgΔz - (1/2)ρΔ(v2) where ΔP = P2 – P1, Δz = z2 – z1, and Δ(v2) = v2

2 – v12. The result above

can be derived from energy considerations, as shown below. Consider an ideal fluid (incompressible, irrotational, and viscous free) in the tube at right. The same mass of fluid will pass through a pipe with different cross sectional areas (e.g. A1 and A2) for every unit of time:

Δm/Δt = ρΔV1/Δt = ρΔV2/Δt where density = mass/volume. But since volume of a cylinder = area x length, and canceling density:

A1Δx1/Δt = A2Δx2/Δt But since speed = length/time:

Q1=A1v1 = A2v2=Q2 This statement of mass conservation is called “The Equation of Continuity” and is frequently given the name "Flow Rate" and symbol "Q=Av". Recall from the first law of thermodynamics that energy of the fluid in the pipe can only change if external forces do work on the fluid, or the fluid particles transfer energy through radiation or heat. For an insulated pipe only work is done on/by the fluid at either end of the pipe:

Wfluid = W1 + W2 = F1Δx1 - F2Δx2 Where the minus sign results from the force being opposite the displacement at the end of the pipe. From the first law of thermodynamics:

Wfluid = ΔE = ΔEk + ΔEg = (1/2)mv22 - (1/2)mv1

2 + mgy2 - mgy1 Recall that m = ρV = ρAΔx and substituting the result for the work done by the fluid:

F1Δx1 - F2Δx2 = (1/2)ρVv22 - (1/2)ρVv1

2 + ρVgy2 - ρVgy1

Dividing by volume and recognizing that Δx/V = 1/A and that F/A = P: P1 – P2 = (1/2)ρv2

2 - (1/2)ρv12 + ρgy2 - ρgy1

But since ΔP = P2-P1 and Δ(v2) = v22 – v1

2 and Δy = y2 – y1: ΔP = -(1/2)ρΔ(v2) - ρgΔy

Q.E.D.

A1

A2 Δx1

Δx2

v1

v2 F2

F1

y1 y2

v2(m/s)2

105

500

P(Pa)

0 v(m/s)

P(Pa)

0


(a) Examples 1) Explain how the equation of continuity describes the motion of river water at a restriction in the

river, for example when boulders block the river path and confine the water passage. Answer: Let A1 be the larger cross sectional area of the river where the water moves slowly. Let A2 be the cross sectional area of the water in the restriction such that A1>A2. Since A1v1 = A2v2 thus v2 = v1(A1/A2) and since A1 > A2 then (A1/A2) > 1 so v2 > v1. 2) Predict what happens to the pressure of the water in the restriction and then use Bernoulli’s

equation to confirm your prediction. Answer: The pressure goes down in the restriction and is largest outside of the restriction. Though slightly surprising at first, think of the restriction as a relief valve – the pressure drops when the fluid is allowed to flow more quickly. Bernoulli’s relationship quickly confirms this prediction:

ΔP = -(1/2)ρΔ(v2) since v2 > v1 then Δ(v2) > 0, and ΔP = P2 – P1 then:

P2 = P1 - (1/2)ρΔ(v2) < P1

Operational Definitions: Ideal Fluids: Conservation of mass (flow rate): Bernoulli's Equation:

- +

- + - +

P2, v2

P1, v1


Conceptual Worksheet C: Dynamic Pressure 1) Arnold Arons: On a windy day, when the wind comes into an open window of your room, the

curtains are blown inward. Occasionally, however, a gust of wind which does not come through the window makes the curtains go out through the window (or up against the screen). This occurs even though there are no other windows open in the room (so the wind isn't blowing through the house in the opposite direction). Explain what could account for this behavior. Can you use the same reasoning to explain why a roof gets ripped off a house during tornadoes and hurricanes? Please use force diagrams in addition to a written explanation.

2) Arnold Arons: When blood flows from the arteries into the capillaries, it goes from one large "tube"

to several much smaller ones. It is well-known that the blood slows down when it reaches the capillaries (a good thing, too--this fact improves the O2 and CO2 exchange that takes place between the blood and the tissues when the blood is in the capillaries). We are used to seeing fluids speed up when the tube is narrowed (like in the case of a nozzle). How can we explain why blood slows down when it enters the capillaries? Assume the flow is free of resistance. Please support your answer with diagrams. For those of you who have had AP&P this answer should be automatic.

vwind

vwind


3) Explain the purpose of “spoiler” on the back of a sports car. Please include a force-diagram on the car in the absence and presence of the spoiler to support your answer.

4) Arnold Arons: The continuity equation for fluid flow says:

a) The velocity of the fluid must be the same in all regions of a pipe. b) The velocity of the fluid increases as the area of a pipe increases. c) The velocity of the fluid decreases as the area of a pipe increases. d) Nothing about the relative velocity of fluids in a pipe.

5) Which force diagram below CORRECTLY describes Bernoulli’s relationship: 6) For those answers that INCORRECTLY explain the force described above, correctly indicate the

direction of the force for the air velocity shown in the space below.

Flift

vair

b) Airplane wing cross

section

Flift

d) Floating ball trick

vair Flift

vair

c) Roof

v=0

Flift

e)

vwater

Shower curtain in a cold shower

v=0

Flift

a)

vair

Closed room, with curtain at open

window.

v=0

Wind into plane of paper.

spoiler

a) b) c) d) e)


7) An aneurysm is a sudden abnormal enlargement of a section of an artery due to a weakening of the arterial wall. a) If the blood flow rate remains constant through the artery, how does the pressure in the region

of enlarged cross section change compared to the pressure on either side of the aneurysm?

b) Diagram your answer. Neglect viscosity of the blood for this answer. 8) Water is flowing as shown in the tube at right (ρ = 1000kg/m3).

a) Determine the pressure difference between top and bottom and the absolute pressure at the bottom if Ptop = 1 atm.

b) What is the ratio of the areas between the top and the bottom of the tubes

and the radii between the top and bottom?

v2=10m/s z2=15m

v1=20m/s z1=5m


Basic Worksheet B: Dynamic Pressure 1) A volleyball player serves up a “round house”, a service in which a

“wicked” topspin is imparted on the volleyball. Correctly force-diagram the spinning volleyball (side view at right) traveling through the air in the direction shown and explain from your diagram if the ball travels farther or shorter than would be expected without spin.

2) Wind traveling parallel to the ground and passing over a building chimney flue may (circle all that

apply and support your choices with a force diagram. a) …decrease the pressure inside the building OR create a draft up the

chimney. b) …increase the pressure inside the building. c) …reduce pressure above the chimney so that smoke is pushed out of

the building. d) …blow smoke into the building. e) …”suck” (pull) warm air out of the building.

3) Determine the math model and density of the medium taken in the

data at right.

1.0

ΔP (kPa)

v2 (m/s)2 100

Pin>Pout

Fv

vwind


4) The Bernoulli principle can be used to "bail out a boat." The drain plug on

the bottom of the hull is removed and “bilge” water (ρ = 1000 kg/m3) will drain out if boat is moving fast enough. The data at right represents the absolute pressure in the bottom of a boat moving at different speeds. Pref = 101000Pa = atmospheric pressure.

a) The linear math model for this data is:

EXPLAIN YOUR ANSWER:

b) The intercept represents: EXPLAIN YOUR ANSWER:

c) Determine the depth of the drain hole below the outside water level when the boat is standing still (g=10 N/kg). EXPLAIN YOUR ANSWER.

d) Determine the minimum speed of the boat needed to keep water from flooding into the open drain plug. EXPLAIN YOUR ANSWER.

v(m/s) P(Pa) 0 102000 4 94000 8 70000 16 -26000 20 -98000

Drain Plug

v


5) Arnold Arons: Water moves through the wider part of this pipe at 4

m/s at a pressure of 2.5x105 Pa. The pipe narrows to half its original diameter. Find (a) the speed and (b) the pressure of the water in the narrower part of the pipe. Assume frictionless flow.

6) Air (indicated by straight arrows) is rushing by a spinning ball with

rotation as shown. a) Draw a microscopic picture showing how the air molecules

build up around the ball in the moving air.

b) Draw the direction of the force on the ball due to its spin in the moving air explaining your answer below.

c) If the pressure difference is 100 Pa across the ball and the density of air is 1.2kg/m3, and the vspin = 0.025v, what is the speed of the air?

d) What league is the pitcher in?

ω


Advanced Worksheet A: Fluid Dynamics 1) A topspin is used to cause a ball to “drop” prematurely in air as predicted by the rules of projectile

motion. A backspin will keep the object in the air longer. a) Explain how this works using a force diagram.

b) Estimate the horizontal pressure difference based on the horizontal force and cross sectional surface area of the baseball.

c) If a ping pong ball (m=10g, diameter =3cm) is struck with sufficient backspin it can travel horizontally, seeming to “defy gravity”. Determine the lift force, and estimate the tangential speed of the outer surface of the ping pong ball necessary to keep the ball traveling horizontally if its linear velocity is 10 m/s. You will need the density of air (web or library) and to carefully examine what Δ(v2) means in terms of the linear motion of the ball and its spin.

d) Now assume that the same tangential speed but as a topspin. Determine the acceleration of the

ball at the instant the ball is horizontal.


2) The 3.0 cm-diameter water line in the figure splits into two 1.0 cm-diameter pipes. All pipes are circular and at the same elevation. At point A, the water speed is 2.0 m/s and the gauge pressure is 50 kPa. What is the gauge pressure at point B?

3) Arteriosclerotic plaques forming on the inner walls of arteries can decrease the effective cross-

sectional area of an artery. Even small changes in the effective area of an artery can lead to very large changes in the blood pressure in the artery and possibly to the collapse of the blood vessel. Imagine a healthy artery, with blood flow velocity of vo=0.14m/s and mass per unit volume of ρ = 1050kg/m3. This leads to a value for the kinetic energy per unit volume of blood of (1/2)ρvo

2 = 10Pa. a) Imagine that plaque has narrowed an artery to one-fifth of its normal cross-sectional area (an

80% blockage). Compared to normal blood flow velocity, vo, what is the velocity of blood as it passes through this blockage?

b) By what factor does the kinetic energy per unit of blood volume change as the blood passes through this blockage?

c) As the blood passes through this blockage, how much does the blood pressure increase or decrease by?


d) For parts a - f imagine that plaque has grown to a 90% blockage. Relative to its initial, healthy

state, by what factor does the velocity of blood increase as the blood passes through this blockage?

e) By what factor does the kinetic energy per unit of blood volume increase as the blood passes through this blockage?

f) What is the magnitude of the drop in blood pressure, ΔP, as the blood passes through this blockage? Use 10 Pa as the normal (i.e., unblocked) kinetic energy per unit volume of the blood.

4) When an open bottle of water is allowed to leak through a hole in

its side (e.g. right), the flow rate (Q=ΔV/Δt) is found to behave as indicated by the following graph. a) Write the equation describe how Q depends on time:

b) At what time is the bottle 73% empty?

c) How much water has emptied out of the container in one minute?

Δz

Δx ΔZ


Fg A

B

C

Unit18: - Bernoulli Deployment Hydrostatic pressure in a column of fluid can be used to force the fluid out of the column. For example, serious lacerations should be elevated to reduce blood loss. Objectives: To provide diagrammatic, graphical, written and mathematical solutions for water, under hydrostatic pressure due to height Δy, being pushed out a hole at a height ΔY from the floor in a bottle and striking a target and length Δx away from the hole (diagram at right). 1) Identify and label the forces on three identical Js both at the top

AND the bottom of each J at the three locations shown. You may ignore the buoyant force due to air at points A and C. The weight of one J is included for reference.

ΣFA = ΣFB = ΣFC = 2) Use the Bernoulli Equation to determine the speed of "vhole" the water at point "C" above. To solve

the problem you are given the following hints: a) The container is open and at atmospheric pressure. b) What pressure is the J at point "C"? This allows you to determine the pressure difference. c) Use the equation of continuity and the fact the surface area at point A is much, much larger than

the surface area of the hole at C to decide what to do with the speed of the water in the Bernoulli equation at point A.

Δψ

Δx ΔY


3) This experiment is repeated in a space lab at one atmosphere pressure on the surface of the moon. In clear, concise sentences write one paragraph explaining how you can use the equation from above to determine Δx.

4) Calculate Δx on the moon. 5) Use your results above and appropriate measurements to determine the location of the target (Δx)

on the floor.

Measured Δy: ________ Measured ΔY: ________ Predicted speed of water at point "C":________TOF:________ Predicted Δx: ________ Actual Δx: ________

U19-Multiparticle Model: Real Fluid Flow

Unit 19: Real Fluids By the time you finish all lab and class activities, you should be able to: 1) Real fluids, e.g. air, can be compressed at sufficiently high pressures appropriate to the

particular fluid. 2) Real fluids have friction between molecules characterized by a dynamic viscosity "η":

€

η ≡FshearΔy

AV

Where Fshear is the force parallel to a layer of fluid of area "A" with multiple layer thickness "Δz" moving at speed "v".

3) Real fluids can be subject to turbulent motion that (in the specific case of a fluid in a tube of radius "r") is characterized by the Reynolds Number:

€

Re ≡ρv2rη

Where "ρ" is the density of the fluid moving at speed "v" and a dynamic viscosity "η". 4) Real fluids have layers that travel at different speeds are can be characterized as being

rotational. 5) In general fluids exert "drag" on moving particles that depend upon the speed "v" of moving

particle, or conversely if the particle is standing still in its own reference frame, the speed of the fluid going by the particle:

Fd = av + bv2 a) That is the drag force depends both linearly and as the square of the speed. At low

speeds for a spherical particle the "viscous" drag force predominates (Re ≤ 0):

€

Fdv ≡ 6πηrv Where "η" is the dynamic viscosity and "r" is the radius of the sphere.

b) At higher speeds the "pressure" drag dominates (Re ≥ 2000):

€

Fdp ≡12ρv2 ACd

Where "ρ" is the density of the fluid, "v" is the speed of the object, "A" is the object's area interacting with the fluid, and "Cd" is a coefficient related to the geometry of the object. In between both forces can play a roll.

6) Poiseuille's Law empirically determined that flow rate in a pipe of radius "r" and length "L" depends on the pressure gradient "ΔP" and viscosity " η ".

€

Q ≡ΔVΔt

=ΔPπr 4

8Lη


Theory – Real Fluid Interactions

In laboratory we will characterize the dynamic viscosity "η" through a simple experiment. Chrome steel ball bearings of weight "mg" and radius "r" will be dropped into a viscous medium and the terminal velocity "v" will be measured. Because the fluid density is significant compared to that of the ball bearings buoyancy must be considered. The interaction and force diagram are shown below. Terminal velocity is a constant velocity, therefore we expect: ΣF = ma = 0 = Fdv + Fb + Fg = +Fdv + Fb - Fg Fdv + Fb = Fg where

€

Fdv ≡ 6πηrv

€

Fb = mf g = ρ f gVf = ρ f g43πr3

€

Fg = mg = ρgV = ρg43πr3

Inserting into the above equation:

€

6πηrv + ρ f g43πr3 = ρg

43πr3

Rearranging we can solve for the dynamic viscosity:

€

η =2g9

r2(ρ − ρ f )v

Where "ρ" is the density of the ball bearing and "ρf" is the density of the fluid. In lab we will examine how the terminal velocity depends on ball bearing radius, and from the known constants, determine the viscosity of the fluid we are characterizing. As part of the deployment activities you will examine the pressure drag force on coffee filters in order to see how changes in independently controlled variables affect the terminal velocity of the coffee filter from equilibrium. Terminal velocity occurs when the pressure drag force and the weight balance each other:

€

12ρv2 ACd = mg ⇒ vterMinal =

2mgρACd

This result should not be memorized because it is only true IN THE ABSENCE OF OTHER FORCES!!!

ball

fluid earth Fg

Fb

Fdv

Fg Fb Fdv


Operational Definitions: Real Fluids: Pressure Drag: Coefficient of Drag: Viscous Drag: Viscosity: Reynolds Number


Conceptual Worksheet C: Pressure Drag 1) The following exercise asks you to change the terminal velocity of a specific object under

consideration. As part of your answer sketch a molecular model and brief explanation why your solution will increase the terminal velocity of the object a) A piece of cardboard oriented horizontally. Fluid only air. You can only use the

cardboard and it cannot be crumpled up. Make it fall faster.

b) A cubicle cardboard box. Fluid only air. Again, crushing is not an option. Make it fall faster.

c) Dropping a rock. You can do NOTHING to the rock. SLOW IT DOWN.

d) Water bottle rocket. You may add a small (negligible mass) of construction paper to allow the rocket to go faster.

vt

vt

vt


1) The terminal velocity of a peregrine falcon of different masses, on different planets (just to make things interesting), in different fluids (ignoring buoyancy and pressure drag), and

different size birds is given by:

a) Assuming all other terms constant and equal to "1", quantitatively sketch the graphical relationships below. The first one has been done for you.

b) Determine the terminal velocity of a 0.5kg falcon with a streamlined body presenting a 100cm2 area with drag coefficient of 0.1 in air on earth.

0 10 m(kg)

10

v(m/s)

0 100 g(N/kg)

10

v(m/s)

0 104 A(cm2)

10

v(m/s)

0 ρ(kg/m3)

0.1

v(m/s)

103


A despondent beetle, intent on committing insecticide, climbs up the stalk of a milkweed plant. With its legs tucked in, it approximates a sphere, 1 mm in diameter, and has a mass of 0.5 mg. 2) If the beetle jumps what will a plot of its acceleration versus time look like? 3) Rank the magnitudes of difference forces acting on the beetle just BEFORE it reaches

terminal velocity: weight Fg, buoyancy Fb, viscous drag Fdv, and pressure drag Fdp. To accurately estimate this you will need to google the coefficient of drag for a sphere and the viscosity of air. Cdsphere = 0.47 ηair = 1.8x10-5 Pl. a) Fg > Fdp > Fdv > Fb b) Fg > Fb > Fdp > Fdv c) Fg = Fdp > Fdv > Fb d) Fg > Fdp > Fb > Fdv e) Fg > Fdv > Fdp > Fb

4) If the beetle jumps, how fast will it fall when it achieves terminal velocity? Is it likely to be

doing a truly terminal act? a) 4.5m/s, no b) 29 m/s, yes probably c) 0.00045 m/s, no d) 21 m/s, yes probably e) 140 m/s, yes definitely

5) If the beetle loses its nerve, clips the fluff from a ripe milkweed seed, and uses it to parachute

down, how fast will it fall? The fluff has a Stokes (as if a sphere obeying Stokes' law) radius of 20 mm and a negligible mass. a) 3.0 m/s b) 1500 m/s c) 0.003 m/s d) 0.0015 m/s e) 1.5 m/s

B) A) C) D) E)

t 0

a

t 0

a

t 0

a

t 0

a

t 0

a


Basic Worksheet B: Viscous Drag 1) Calculate the viscosity of honey from the following experiment. A butter knife with 25cm2

area drags a 5mm thick layer of honey across a very flat piece of toast. The force required to pull the knife is exerted by the mass attached by a string. It takes 5 seconds for the square knife to travel the length of the blade.

a) Interaction and force diagram the forces on the blade.

b) Solve for the speed of the blade.

c) Determine the shear force on the honey.

d) Calculate the viscosity of the honey in Poiseuille (Pl), poise (P) and centipoises (cp). Compare it with an answer you find on-line. Is your answer reasonable?

e) Honey's viscosity is temperature dependent, as is its density. If viscosity measures the making and breaking of bonds, explain why increased temperature decreases the viscosity of honey.

100g


2) A 4.0 mL syringe has an inner diameter of 4.0 mm, with an 18 gauge (about 1mm) needle, and a plunger pad diameter (where you place your finger) of 1.2 cm. A nurse uses the syringe to inject medicine into a patient whose blood pressure is 140/100. The venous pressure is about 2mm Hg. REDO – CRAPPY PROBLEM a) What is the minimum force the nurse needs to apply to the syringe?

b) The nurse empties the syringe in 3.6s. What is the flow speed of the medicine through the needle?

3) Estimate the speed of blood through an artery (r=5mm) in your arm if the Reynolds number

is about 2000, ρ=1000kg/m3 and viscosity of 0.003Pl. 4) If the blood flow rate in the artery above is 5L/min, calculate the pressure gradient using

Poiseuille's Law over 1cm radius of the artery. a) First solve this problem algebraically, i.e. determine ΔP/L as a function of flow rate,

radius, and viscosity.

b) Then plug in numbers to determine the pressure gradient.


5) Graph flow rate out of a ketchup bottle as a function of cap radius "r" and neck length "L". The constants are below each graph.

An insect trap consists of deep jar of alcohol. Spherical beetles climb in and sink to the bottom at constant speed where they are trapped and expire. The beetle size is about 1cm, vt=0.5m/s, ρ=900kg/m3, and the viscosity of alcohol is about 0.001Pas. Also: 6) A beetle sinks because…

a) A beetle has greater density than the oil. b) The oil makes and breaks bonds with a beetle. c) A beetle has greater weight than the oil. d) A beetle's buoyant force is greater than its weight e) The oil makes insufficient collisions with the beetle to keep it afloat.

6) Select the force diagram that correctly depicts the forces on the beetle at terminal velocity. Key: Fg = weight, Fb = buoyancy, Fv = Bernoulli force, Fdp = pressure drag, Fdv = viscous drag. Forces are to scale.

7) Use the correct force diagram to determine the algebraic expression for the terminal velocity of the object described above.

0 5 L(cm)

.03

ΔV/Δt

0 r(cm)

.02

ΔV/Δt

1

ΔP=1000Pa, L=0.05m, η=50Pl ΔP=1000Pa, r=0.01m, η =50Pl

b) a) c) d) e)

Fg Fdp

Fb

Fg

Fdp

Fg

Fb Fdv

Fg Fdv

Fb

Fg

Fdp Fb

B) A) C) D) E)


Advanced Worksheet A: Pressure and Viscous Drag 1) In a desperate attempt to hide treasure from the allied forces in World War II, the Nazis

dropped gold bullion into deep alpine lakes.

a) Interaction and force diagram all the forces acting on a solid gold cannonball as it sinks in water. The four forces should add up to zero when you have finished drawing your force diagram in the grid at right.

b) Rank the forces from largest to smallest explaining your prediction.

c) Write down algebraic equations for each force below (weight, buoyancy, viscous drag and pressure drag). Information you will need: area of hemisphere 2πr2 (m2) and volume

of a sphere

€

Vsphere =43πr3 (m3).

d) At terminal velocity ΣF=0 for the cannonball. Set up a quadratic equation to solve for the velocity "v". A quadratic equation is of the form av2 + bv + c = 0

e) Solve for the value of each force assuming r = 10cm and the terminal velocity is 7.1m/s.

Write down your answer in the following format (weight is done for you). You will need the density of water "ρ" (kg/m3), viscosity "η" of water (Pl), and coefficient of drag "Cd" (unitless) for a sphere (find using web search).

f) Rank the forces. Were your predictions correct? Which force (after weight) was most important? Could you have simplified your expression for terminal velocity?


2) A hemispherical parachute made of an air-impervious material is 8.0 m in diameter. The mass of parachute and load combined is 80 kg. (Assume a drag coefficient of 1.42 based on frontal area facing flow and the viscosity of air is 1.8x10-5Pl) a) How fast will the parachute descend in still air assuming pressure drag dominates?

b) How fast will the parachute descend in still air assuming viscous drag dominates?

c) Calculate the Reynolds number based on these two calculations and determine whether pressure drag or viscous drag dominates:

d) How far would an object with no drag have to fall to reach this same speed as it accelerates from rest? (Landing with a parachute will thus involve about the same impact as would a downward free jump of this distance.)

3) Trees and vines have pipes ("xylem") that convey water upward from roots to leaves. Let us

assume that these pipes are cylindrical in cross section, that they run vertically, and that they are functionally uninterrupted. a) Assume that as water ascends due to the force from root pressure "Fp" the loss of pressure

due to change in gravitational force "Fg" is just equal to that due to pipe resistance "Fdv" (the latter according to Hagen-Poiseuille's Law). That is, the total pressure drop will be half gravitational and half viscous. Which set of force diagrams accurately depicts the forces acting on water molecules as they are pumped upwards through the xylem?

b) What would be the average speed of ascent in the largest vessels, 0.5 mm in diameter, as found in some vines?

b) a) c) d)

Fg Fdv

Fp

Fg

Fdv

Fg

Fp Fdv

Fg Fdv

Fp


c) What would be the average speed where the vessels are 0.05 mm in diameter, as in some conifers?

d) What pressure difference (negative, of course) would be needed to raise water to a height of 40 meters in such conifers?

4) A sea jelly (a.k.a. jellyfish) of diameter 10cm expels saltwater through contraction of its body

(umbrella) and moves in spurts at an average speed of 1m/s. c) Calculate the Reynolds number of a sea jelly and determine what form of drag is most

important.

d) Look at the following video: http://vodpod.com/watch/3987024-visualizing-jellyfish-motion, what evidence from the videos supports this conclusion?

e) Support your answers by calculating the ratio of the pressure to viscous drag assuming the sea jelly is approximately a sphere.


Unit 19: Drag Force Deployment Pressure drag plays enormous roles in perambulation from the ejection of microscopic dust pollen to whales swimming. Objectives: To provide diagrammatic, graphical, written and mathematical solutions for the coefficient of drag in air for a helium filled balloon at terminal velocity. 1) Sketch a kinematics stack for an object released from rest to some time "tt" after it has

reached terminal velocity. Explain your answer below. Hint: "think exponentially"

8) Draw a force diagram on the object at terminal velocity to scale in the

grid below assuming buoyancy is three times its weight. In the space at left write a mathematical statement for these forces solving for the pressure drag ONLY in terms the mass of the balloon and the gravitational field.

a) Generic force equation using labels from your diagram

at right:

b) Specific solution: Fdp =

x

t v

t

a

t

tt


9) Based on your knowledge of pressure drag (for example from class notes and readings), predict what the graph of terminal velocity versus mass will look like in the graph at right. Explain your answer below.

10) From the measured terminal velocity "vt" for the object of measured mass "m" and area "A",

calculate the coefficient of drag of the object, Cd, and compare it for a tear drop shape found online.

v(m/s)

m(kg)

four modeling instruction fluids modules

Documents