lecture 23 temperature and heat. bernoulli’s equation the general case, where both height and...

Lecture 23

Temperature and Heat

Bernoulli’s Equation

The general case, where both height and speed may change, is described by Bernoulli’s equation:

This equation is essentially a statement of conservation of energy in a fluid.

A person’s blood pressure is generally measured on the arm, at approximately the same level as the heart. How would the results differ if the measurement were made on the person’s leg instead?

a) blood pressure would be lower

b) blood pressure would not change

c) blood pressure would be higher

Blood PressureBlood Pressure

A person’s blood pressure is generally measured on the arm, at approximately the same level as the heart. How would the results differ if the measurement were made on the person’s leg instead?

a) blood pressure would be lower

b) blood pressure would not change

c) blood pressure would be higher

Assuming that the flow speed of the blood does not change,

then Bernoulli’s equation indicates that at a lower height, the

pressure will be greater.

Blood PressureBlood Pressure

Fluid Flow Fluid Flow

Water flows through a 1-cm diameter pipe

connected to a ½-cm diameter pipe, both

horizontal. Compared to the speed of the

water in the 1-cm pipe, the speed in the

½ -cm pipe is:

a) one-quarter as fast

b) one-half as fast

c) the same

d) twice as fast

e) four times as fast

The area of the small pipe is less, so we know that the water will flow

faster there. Because AA ∝∝ rr22, when the radius is reduced byradius is reduced by one-halfone-half,

the area is reduced by one-quarterarea is reduced by one-quarter, so the speed must increase by speed must increase by

four timesfour times to keep the flow rate ((AA vv)) constant.

Fluid Flow Fluid Flow

a) one-quarter as fast

b) one-half as fast

c) the same

d) twice as fast

e) four times as fast

Water flows through a 1-cm diameter

pipe connected to a ½-cm diameter pipe.

Compared to the speed of the water in

the 1-cm pipe, the speed in the ½ -cm

pipe is:

v1 v2

A tank of water filled to a depth d has a hole in its side a height h above the table on which is rests. Show that water emerging from the hole hits the table at a horizontal distance of from the base of the tank.

Heat

Definition of heat:

Heat is the energy transferred between objects because of a temperature difference.

Objects are in thermal contact if heat can flow between

them.

When the transfer of heat between objects in thermal contact ceases, they are in thermal equilibrium.

The Zeroth Law of ThermodynamicsIf object A is in thermal equilibrium with object B, and object C is also in thermal equilibrium with object B, then objects A and C will be in thermal equilibrium if brought into thermal contact.

That is, temperature is the only factor that determines whether two objects in thermal contact are in thermal equilibrium or not.

Object B can then be a thermometer, providing a scale to compare objects

Length of a metal rodVolume of a gas held at constant pressurePressure of a gas held at constant volumeElectrical resistance of a metal conductorVolume of a liquid

A good thermometric property should to be linear with temperature over a wide range of temperature.

Thermometric properties

Common Temperature ScalesThe Celsius scale:Water freezes at 0° Celsius.Water boils at 100° Celsius.

The Fahrenheit scale:Water freezes at 32° Fahrenheit.Water boils at 212° Fahrenheit.

Andreas Celsius: 1701-1744, Swedish astronomer

Daniel Fahrenheit: 1686-1736, German physicist

Copyright © 2010 Pearson Education, Inc.

A natural zero point for temperature

The pressure in a gas is proportional to its temperature.

The proportionality constant is different for different gases, but they all reach zero pressure at the same temperature, which we call absolute zero

Absolute Zero = -273.15 oC


Temperature Scales

The Kelvin scale is similar to the Celsius scale, except that the Kelvin scale has its zero at absolute zero.

Thermal ExpansionMost substances expand when heated; the change in length or volume is typically proportional to the change in temperature.

The proportionality constant is called the coefficient of linear expansion.

Some typical coefficients of thermal expansion

Invar ~1.0 x 10-6

a nickel steel alloy (65% Fe +35%) Ni noted for its very small expansion.Charles Guillaume received the Nobel Prize in Physics in 1920 for the invention

http://en.wikipedia.org/wiki/Nickel

http://en.wikipedia.org/wiki/Steel

http://en.wikipedia.org/wiki/Steel

http://en.wikipedia.org/wiki/Alloy

Thermal Expansion of a bi-metallic strip

A bimetallic strip consists of two metals of different coefficients of thermal expansion, A and B in the figure. It will bend when heated or cooled.

Thermal Expansion

The expansion of an area of a flat substance is derived from the linear expansion in both directions

a) gets larger

b) gets smaller

c) stays the same

d) vanishes

Metals such as brass expand when heated. The thin brass plate in the movie has a circular hole in its center. When the plate is heated, what will happen to the hole?

Steel ExpansionSteel Expansion

a) gets larger

b) gets smaller

c) stays the same

d) vanishes

Imagine drawing a circle on the plate. This circle will expand This circle will expand outward along with the rest of the outward along with the rest of the plate.plate. Now replace the circle with the hole, and you can see that the hole will expand outward as well. Note that the material does Note that the material does NOTNOT “expand inward” to fill the hole!!“expand inward” to fill the hole!!

expansionexpansion

Metals such as brass expand when heated. The thin brass plate in the movie has a circular hole in its center. When the plate is heated, what will happen to the hole?

Steel ExpansionSteel Expansion

Thermal Volume Expansion

The change in volume of a solid is also derived from the linear expansion:

For liquids and gases, only the coefficient of volume expansion is defined:

Some typical coefficients of volume expansion

Pyrex Glass ~1.0 x 10-5

Thermal Expansion of Water

Water also expands when it is heated, except when it is close to freezing; it actually expands when cooling from 4° C to 0° C. This is why ice floats and frozen bottles burst.

GlassesGlasses

a) run hot water over them both

b) put hot water in the inner one

c) run hot water over the outer one

d) run cold water over them both

e) break the glasses

Two drinking glasses are

stuck, one inside the

other. How would you get

them unstuck?

a) run hot water over them both

b) put hot water in the inner one

c) run hot water over the outer

one

d) run cold water over them both

e) break the glasses

Running hot water over only the outer glassouter glass will

allow the outer one to expandouter one to expand, while the inner glass

remains relatively unchanged. This should loosen

the outer glass and free it.

GlassesGlasses

Two drinking glasses are

stuck, one inside the

other. How would you

get them unstuck?

A grandfather clock uses a brass pendulum to keep perfect time at room temperature. If the air conditioning breaks down on a very hot summer day, how will the grandfather clock be affected?

a) clock will run slower than usual

b) clock will still keep perfect time

c) clock will run faster than usual

Grandfather ClockGrandfather Clock

A grandfather clock uses a brass pendulum to keep perfect time at room temperature. If the air conditioning breaks down on a very hot summer day, how will the grandfather clock be affected?

a) clock will run slower than usual

b) clock will still keep perfect time

c) clock will run faster than usual

The pendulum will expand, so its length will increase. The period of a pendulum depends on the length, as shown below, so the period will also increase. Thus, the clock will run slow.

Grandfather ClockGrandfather Clock

Follow-upFollow-up: Roughly how much slower will it run: Roughly how much slower will it run ??

g

LT2π


Heat and Mechanical Work

Heat is another form of energy.James Joule used a device similar to

this one to measure the mechanical equivalent of heat:

One kilocalorie (kcal) is defined as the amount of heat needed to raise the temperature of 1 kg of water from 14.5° C to

15.5° C.


Heat Capacity

The heat capacity of an object is the amount of heat added to it divided by its rise in temperature:

Q is positive if ΔT is positive; that is, if heat is added to a system.

Q is negative if ΔT is negative; that is, if heat is removed from a system.


Specific Heat

The heat capacity of an object depends on its mass and on a property of the material itself: the specific heat

“heat capacity per kilogram”


Specific heats of various materials

A ceramic coffee cup, with c=1090, and m =116g, is initially at room temperature (24.0 °C). If 225 g of 80.3 °C coffee and 12.2 g of 5.00 °C cream are added to the cup, what is the equilibrium temperature of the system? Assume that no heat is exchanged with the surroundings, and that the specific heat of coffee and cream are the same as the specific heat of water.

Two Liquids Two Liquids

a) the cooler one

b) the hotter one

c) both the same

Two equal-mass liquids, initially at the

same temperature, are heated for the

same time over the same stove. You

measure the temperatures and find that

one liquid has a higher temperature than

the other. Which liquid has a higher

specific heat?

Both liquids had the same increase in internal energy,

because the same heat was added. But the cooler liquidcooler liquid

had a lower temperaturelower temperature change.

Because QQ = = mcmcΔΔTT, if QQ and mm are both the same and ΔΔTT

is smaller, then cc (specific heat) must be bigger.

Two Liquids Two Liquids

a) the cooler one

b) the hotter one

c) both the same

Two equal-mass liquids, initially at the

same temperature, are heated for the

same time over the same stove. You

measure the temperatures and find that

one liquid has a higher temperature than

the other. Which liquid has a higher

specific heat?

Thermal equilibrium is reached by means of thermal contact, which in turn can occur through three different mechanisms

Heat Transfer Mechanisms

conduction : it occurs when objects at different temperature are in physical contact (e.g. when holding a hot potato). Faster moving molecules in the hotter object transfer some of their energy to the colder one

convection : this occurs mainly in fluids. In a pot of water on a stove, the liquid at the bottom is heated by conduction. The hot water has lower density and rises to the top, cold water from the top falls to the bottom and gets heated, etc.

radiation : any object at non-zero temperature emits radiation (in the form of electromagnetic waves). The effect is more noticeable when standing next to a red-hot coal fire, or in the sun rays

ConductionConduction is the flow of heat directly through a

physical material

The amount of heat Q that flows through a rod:• increases proportionally to the cross-sectional area A• increases proportionally to ΔT from one end to the other• increases steadily with time• decreases inversely with the length of the rod

The constant k is called the thermal conductivity of the material

Some Typical Thermal Conductivities

Substances with high thermal conductivities are good conductors of heat; those with low thermal conductivities are good insulators.

Two metal rods—one lead, the other copper—are connected in series, as shown. Note that each rod is 0.525 m in length and has a square cross section 1.50 cm on a side. The temperature at the lead end of the rods is 2.00°C; the temperature at the copper end is 106°C. (a) The average temperature of the two ends is 54.0°C. Is the temperature in the middle, at the lead-copper interface, greater than, less than, or equal to 54.0°C? Explain. (b) find the temperature at the lead-copper interface.

kPb = 34.3 W / (kg-m)kCu = 395 W / (kg-m)


Assumptions: •The end points are infinite heat reservoirs... so their temperature doesn’t change for this exercise •The temperature is constant in time at every point. This is not true at moment of thermal connection. We are solving the “steady state” condition, when the temperature at each point doesn’t change.

kPb = 34.3 W / (kg-m)kCu = 395 W / (kg-m)


- The heat (per unit time) through the lead must equal that through the copper

- The lead has a smaller thermal conductivity than the copper

The lead requires a larger temperature difference across it than the copper, to get the same heat flow. So TJ > 54o C

kPb = 34.3 W / (kg-m)kCu = 395 W / (kg-m)

(a)


kPb = 34.3 W / (kg-m)kCu = 395 W / (kg-m)

(b)

Objects that are hot enough will glow – first red, then yellow, white, and blue.

RadiationAll objects give off energy in the form of radiation, as electromagnetic waves (light) – infrared, visible light, ultraviolet – which, unlike conduction and convection, can transport heat through a vacuum.

ConvectionConvection is the flow of fluid due to a difference in temperatures, such as warm air rising. The fluid “carries” the heat with it as it moves.

Objects at body temperature radiate in the infrared, and can be seen with IR night vision optics.

Radiation - even if you can’t see it

RadiationThe amount of energy radiated by an object due to its temperature is proportional to its surface area and also to the fourth (!) power of its temperature.

It also depends on the emissivity, which is a number between 0 and 1 that indicates how effective a radiator the object is; a perfect radiator would have an emissivity of 1.

Here, e is the emissivity, and σ is the Stefan-Boltzmann constant:

The surface of the Sun has a temperature of 5500 oC. (a) Treating the Sun as a perfect blackbody, with an emissivity of 1.0, find the power that it radiates into space. The radius of the sun is 7.0x108 m, and the temperature of space can be taken to be 3.0 K (b) the solar constant is the number of watts of sunlight power falling on a square meter of the Earth’s upper atmosphere. Use your result from part (a) to calculate the solar constant, given that the distance from the Sun to the Earth is 1.5x1011 m.

The surface of the Sun has a temperature of 5500 oC. (a) Treating the Sun as a perfect blackbody, with an emissivity of 1.0, find the power that it radiates into space. The radius of the sun is 7.0x108 m, and the temperature of space can be taken to be 3.0 K (b) the solar constant is the number of watts of sunlight power falling on a square meter of the Earth’s upper atmosphere. Use your result from part (a) to calculate the solar constant, given that the distance from the Sun to the Earth is 1.5x1011 m.

emissivity (a)

(b)

Heat Conduction Heat Conduction

Given your experience of

what feels colder when you

walk on it, which of the

surfaces would have the

highest thermal

conductivity?

a) a rugb) a steel surfacec) a concrete floord) has nothing to do with

thermal conductivity

Heat Conduction Heat Conduction

Given your experience of

what feels colder when you

walk on it, which of the

surfaces would have the

highest thermal

conductivity?

a) a rugb) a steel surfacec) a concrete floord) has nothing to do with

thermal conductivity

The heat flow rate is k A (T1 − T2)/l. All things being

equal, bigger k leads to bigger heat loss.From the book: Steel = 40, Concrete = 0.84,Human tissue = 0.2, Wool = 0.04, in units of J/(s.m.C°).

lecture 23 temperature and heat. bernoulli’s equation the general case, where both height and...

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