the pipe is narrow where point a is located, widens out some where points b and c are located, and...

The pipe is narrow where point A is located, widens out some where points B and C are located, and widens again where point D is located. The level of points C and D is lower than that of points A and B. If the fluid is at rest, rank the points based on their pressure.

1. Equal for all four 2. A>B=C>D 3. D>B=C>A 4. A=B>C=D

Fluid in a pipe – question 1

5. C=D>A=B 6. A>B>C>D 7. D>C>B>A 8. It's ambiguous - there are

two possible answers

The pipe is narrow where point A is located, widens out some where points B and C are located, and widens again where point D is located. The level of points C and D is lower than that of points A and B. If the fluid is flowing from left to right, rank the points based on the fluid speed.





The pipe is narrow where point A is located, widens out some where points B and C are located, and widens again where point D is located. The level of points C and D is lower than that of points A and B. If the fluid is flowing from left to right, rank the points based on the pressure.





The pipe is narrow where point A is located, widens out some where points B and C are located, and widens again where point D is located. The level of points C and D is lower than that of points A and B. If the fluid is flowing from right to left, rank the points based on the pressure.





Three holes in the fountain A cylinder that is full of water and open to the atmosphere at the top stands upright on a table. There are three holes in the side of the cylinder, although these are covered to start with. One hole is 1/4 of the way down from the top, while the other two are 1/2 and 3/4 of the way down. When the holes are uncovered, water shoots out. Which hole shoots the water farthest horizontally on the table?

1. The hole closest to the top. 2. The hole halfway down. 3. The hole closest to the bottom. 4. It's a three-way tie.

There’s three holes in my bucket…

Simulation

Start with a hole a distance h from the top of the cylinder, which has a height H. With what speed does the water emerge from the hole?

Let's apply Bernoulli's equation:

Point 2: Outside the cylinder, at the hole.

Point 1: ?

2 21 1 1 2 2 2

1 1

2 2gy v P gy v P


Simulation

Start with a hole a distance h from the top of the cylinder, which has a height H. With what speed does the water emerge from the hole?

Let's apply Bernoulli's equation:


Point 1: One good place is inside the cylinder, at the top.

2 21 1 1 2 2 2

1 1

2 2gy v P gy v P


Both points are exposed to the atmosphere, so

We can define y2 = 0, so y1 = h .Our equation is now:

Cancel factors of density:


Point 1: One good place is inside the cylinder, at the top

2 21 1 1 2 2 2

1 1

2 2gy v P gy v P

1 2 atmP P P

2 21 2

1 1

2 2gh v v

2 21 2

1 1

2 2gh v v


Now bring in continuity:

In this case we can say that A1, the cross-sectional area of the cylinder, is much larger than A2, the area of the hole, so v1 is negligible compared to v2.

Our equation is now:

Amazingly, we get the familiar result:

Maybe this is not so amazing – we got this result in other cases by applying energy conservation.

1 1 2 2Av A v

22

1

2gh v

2 2v gh

2 21 2

1 1

2 2gh v v

Projectile motion

Now, we do a projectile motion analysis to see how far horizontally the water goes.

The remaining vertical distance to the table is H – h, and the water has no initial velocity vertically (it’s all horizontal).

So, the time of flight is

The horizontal distance traveled is:

2( )H ht

g

21

2H h gt

2 2v gh

2( )2 2 ( )x

H hx v t gh h H h

g

x is maximized when h = H/2.

Thermal expansionLinear expansionMost materials expand when heated. As long as the temperature change isn't too large, each dimension of an object experiences a change in length that is proportional to the change in temperature.

or, equivalently,

where L0 is the original length, and is the coefficient of linear expansion, which depends on the material.

0L L T 0 1L L T

Material (× 10-6/°C) Material (× 10-6/°C)

Aluminum 23 Glass 8.5

Copper 17 Iron 12

Thermal expansionVolume expansionFor small temperature changes, we can find the new volume using:

or, equivalently,

where V0 is the original volume.

0 3V V T 0 1 3V V T

Bimetallic strip

A bimetallic strip is made from two different metals that are bonded together. The strip is straight at room temperature, but it curves when it is heated. How does it work?

What is a common application of a bimetallic strip?

Bimetallic strip

A bimetallic strip is made from two different metals that are bonded together. The strip is straight at room temperature, but it curves when it is heated. How does it work?

The metals have equal lengths at room temperature but different expansion coefficients, so they havedifferent lengths when heated.

What is a common application of a bimetallic strip? A bimetallic strip can be used as a switch in a thermostat. When the room is too cool the strip completes a circuit, turning on the furnace. The furnace goes off when the room (and the strip) warms up.

What happens to holes?

When an object is heated and expands, what happens to any holes in the object? Do they get larger or smaller?

1. The holes get smaller

2. The holes stay the same size

3. The holes get larger

Holes expand, too

Holes expand as if they were filled with the surrounding material.

If you draw a circle on a disk and then heat the disk, the whole circle expands.

Removing the material inside the circle before heating produces the same result – the hole expands.

Holes expand, too

Thermal Stress

If an object is heated or cooled and it is not free to expand or contract, the thermal stresses can be large enough to cause damage. This is why bridges have expansion joints (check this out where the BU bridge meets Comm. Ave.). Even sidewalks are built accounting for thermal expansion.

Materials that are subjected to thermal stress can age prematurely. For instance, over the life of a airplane the metal is subjected to thousands of hot/cold cycles that weaken the airplane's structure.

Another common example occurs with water, which expands by 10% when it freezes. If the water is in a container when it freezes, the ice can exert a lot of pressure on the container.

A black can and a white can

Two cans, one black and one white, are at room temperature. They are then exposed to a heat lamp. Which one heats up fastest? The cans are identical except for their surfaces.

1. the black can

2. the white can

3. they heat up at the same rate


We've probably all noticed, by leaving black objects out in the sun, that they heat up fastest. The black can absorbs radiation more efficiently than does the white can, which reflects more of the radiation away.


The same two cans are then filled with hot water. Which cools down fastest?

1. the black can

2. the white can

3. they cool down at the same rate

Heat transfer

Heat naturally flows from higher-temperature regions to lower-temperature regions.

The three basic mechanisms by which heat is transferred are conduction, convection, and radiation. We'll look at each of these separately, but in a given situation more than one mechanism might be important.

Conduction

Thermal conduction involves energy in the form of heat being transferred from a hot region to a cooler region through a material. At the hotter end, the atoms, molecules, and electrons vibrate with more energy than they do at the cooler end. The atoms, molecules, and electrons don't flow from one place to the other - the energy flows through the material, passed along by the vibrations.

Conduction

The rate at which heat is conducted along a bar of length L depends on the length, the cross-sectional area A, the temperature difference between the hot and cold ends, TH - TC, and the thermal conductivity k of the material.

The rate of energy transfer is power, so:

( )conducted H CP Q t k AL T T

Thermal conductivity

Metals generally have high thermal conductivities because of the free electrons that move around randomly. These are very efficient at transferring energy through the metal. Copper, for instance, has a thermal conductivity of 400 W/(m K), compared to 0.024 W/(m K) for foam insulation.

R values

Insulating materials are rated in terms of their R values, which measures their resistance to conduction. The higher the R, the lower the conductivity.

In terms of the thickness, L:L

Rk

A conduction sandwichA typical conduction problem involves creating a sandwich of two (or more) layers and determining the temperature at the interface(s) between the layers.

Consider a two-layer problem where one layer has twice the thickness and six times the thermal conductivity as the other layer, but the layers have the same area.

A conduction sandwichTo find the temperature at the interface between the layers (after thermal equilibrium has been reached) you should:

1. find that the unknown temperature is halfway between the temperature on one side and the temperature on the other side (T = ?)

2. set up a ratio where the change in temperature across a layer is proportional to the thickness of the layer (T = ?)

3. set up a ratio where the change in temperature across a layer is inversely proportional to the thickness of the layer (T = ?)

4. set the rate of heat flow through one layer equal to

the rate of heat flow through the other layer (T = ?)

A conduction sandwichTo find the temperature at the interface between the layers (after thermal equilibrium has been reached) you should:

1. find that the unknown temperature is halfway between the temperature on one side and the temperature on the other side (T = 12° C)

2. set up a ratio where the change in temperature across a layer is proportional to the thickness of the layer (T = 8° C )

3. set up a ratio where the change in temperature across a layer is inversely proportional to the thickness of the layer (T = 16° C)

4. set the rate of heat flow through one layer equal to

the rate of heat flow through the other layer (T = 18° C )

Convection

Heat transfer in fluids generally takes place via convection, in which flowing fluid carries heat from one place to another. Convection currents are produced by temperature differences. Hotter (less dense) parts of the fluid rise, while cooler (more dense) areas sink. Birds and gliders make use of upward convection currents to rise, and we also rely on convection to remove ground-level pollution.

Forced convection, where the fluid does not flow of its own accord but is pushed, is often used for heating (e.g., forced-air furnaces) or cooling (e.g., fans, automobile cooling systems).

Thermal radiation

Thermal radiation involves energy transferred via electromagnetic waves. Often this is infrared radiation, but it can also be visible light or radiation of higher energy.

Thermal radiation is relatively safe, and is not the dangerous nuclear radiation associated with nuclear bombs, etc.

All objects continually absorb thermal energy and radiate it away again. When everything is at the same temperature, the amount of energy received is equal to the amount given off and no changes in temperature occur. If an object emits more than it absorbs, though, it cools down.

Thermal radiation

For an object with a temperature T (in Kelvin) and a surface area A, the net rate of radiated energy depends strongly on temperature:

where Tenv is the temperature of the surrounding environment, and the Stefan-Boltzmann constant is

σ = 5.67 x 10-8 W/m2

e is the emissivity. It is a measure of how efficiently an object absorbs and emits radiated energy. Highly reflective objects have emissivities close to zero. Black objects have emissivities close to 1. An object with e = 1 is called a perfect blackbody.

4 4( )net radiated absorbed envP P P eA T T

Thermal radiation

The best absorbers are also the best emitters. Black objects heat up faster than shiny ones, but they cool down faster too.

This is exactly what we observe with our black can and white can, as the cans cool.

Newton’s Law of Cooling

In many situations (and our cans are an example of this), the temperature of a hot object that is cooling follows an exponential decay. What does this tell us about what the rate at which the object loses energy? How does this rate of energy loss depend on temperature?


In many situations (and our cans are an example of this), the temperature of a hot object that is cooling follows an exponential decay. What does this tell us about what the rate at which the object loses energy? How does this rate of energy loss depend on temperature?

Exponential decay is characteristic of a rate that is proportional to a quantity, in this case the temperature difference between the object and the surroundings.

h is the heat transfer coefficient

( )object env

dQhA T T

dt


With the rate of energy loss being proportional to the temperature difference, the exponential equation for the temperature of the object as a function of time is

λ is the decay rate, which depends on the surface area and emissivity.

( ) tobject env initial envT T T T e

Heat

What is heat?

Heat

Heat is energy transferred between a system and its surroundings because of a temperature difference between them.

Specific heat

The specific heat of a material is the amount of heat required to raise the temperature of 1 kg of the material by 1°C.

The symbol for specific heat is c.

Heat lost or gained by an object is given by: Q mc T

Material c (J/(kg °C)) Material c (J/(kg °C))

Aluminum 900 Water (gas) 1850

Copper 385 Water (liquid) 4186

Gold 128 Water (ice) 2060

A change of state

Changes of state occur at particular temperatures, so the heat associated with the process is given by:

Freezing or melting:

where Lf is the latent heat of fusion

Boiling or condensing:

where Lv is the latent heat of vaporization

For water the values are:

Lf = 333 kJ/kg

Lv = 2256 kJ/kg c = 4.186 kJ/(kg °C)

fQ mL

vQ mL

Which graph?

Simulation

Heat is being added to a sample of water at a constant rate. The water is initially solid, starts at -10°C, and takes 10 seconds to reach 0°C.

You may find the following data helpful when deciding which graph is correct:

Specific heats for water: cliquid = 1.0 cal/g °C and

cice = csteam = 0.5 cal/g °C Latent heats for water: heat of fusion Lf = 80 cal/g and heat of vaporization Lv = 540 cal/g

Which graph shows correctly the temperature as a function of

time for the first 120 seconds?

Which graph?

Which graph shows correctly the temperature as a function of time for the first 120 seconds?

1. Graph 1

2. Graph 2

3. Graph 3

4. Graph 4

5. Graph 5

6. None of the above

Ice water

100 grams of ice, with a temperature of -10°C, is added to a styrofoam cup of water. The water is initially at +10°C, and has an unknown mass m. If the final temperature of the mixture is 0°C, what is the unknown mass m? Assume that no heat is exchanged with the cup or with the surroundings.

Use these approximate values to determine your answer:

Specific heat of liquid water is about 4000 J/(kg °C) Specific heat of ice is about 2000 J/(kg °C) Latent heat of fusion of water is about 3 x 105 J/kg

Ice water

One possible starting point is to determine what happens if nothing changes phase. How much water at +10°C does it take to bring 100 g of ice at -10°C to 0°C? (The water also ends up at 0°C.)

You can do heat lost = heat gained or the equivalent method:

Plugging in numbers gives:

Lot's of things cancel and we're left with:

100 g = 2m, so m = 50 g. So, that's one possible answer.

J J

(100 g) 2000 10 C 4000 ( 10 C) 0kg°C kg°C

m

0Q

0ice ice ice water water waterm c T m c T

Ice water

Challenge for next time: find the range of possible answers for m, the mass of the water.

Whiteboard

the pipe is narrow where point a is located, widens out some where points b and c are located, and...

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