IB DP Physics: Topic 8. Energy production: 8.2 Thermal energy transfer: Study Notes

8.2 Thermal energy transfer

Essential Idea:
For simplified modelling purposes the Earth can be treated as a black-body radiator and the atmosphere treated as a grey-body.

Understandings:

  • Conduction, convection and thermal radiation
  • Black-body radiation
  • Albedo and emissivity
  • The solar constant
  • The greenhouse effect
  • Energy balance in the Earth surface–atmosphere system

Applications and Skills:

  • Sketching and interpreting graphs showing the variation of intensity with wavelength for bodies emitting thermal radiation at different temperatures
  • Solving problems involving the Stefan–Boltzmann law and Wien’s displacement law
  • Describing the effects of the Earth’s atmosphere on the mean surface temperature
  • Solving problems involving albedo, emissivity, solar constant and the Earth’s average temperature

Data booklet reference:

Big Ideas

        Most energy sources can be traced back the sun, our ultimate primary source

        Energy sources must be compared based on many factors including energy density, cost, availability, politics, safety, and environmental impact

        No energy source can be converted to electricity with 100% efficiency

        All energy sources have advantages and drawbacks and it important to understand the complete picture

        Every object with a temperature above 0 K emits thermal radiation

        Radiation intensity is related to separation distance by the inverse square law (similar to force fields)

        The Earth’s climate relies on a delicate thermal energy balance where total energy in equals total energy out 

Global Energy Usage

Rank

Energy Source

%

1

Oil

32%

2

Coal

28%

3

Natural Gas

22%

4

Biomass

10%

5

Nuclear

5%

6

Hydropower

2.5%

Efficiency

 

Sankey Diagram Rules:

Width of the arrow proportional to the amount of energy

Energy Density

 

Definition

Units

Specific Energy

Energy transferred per unit mass

J kg-1

Energy Density

Energy transferred per unit volume

J m-3

Primary and Secondary Sources

Primary Energy Sources

Secondary Energy Sources

Energy sources found in the natural environment

(fossil fuels, solar, wind, nuclear, hydro, etc.)

Useful transformations of the primary sources

(electricity, pumped storage for hydro, etc.)

Fossil Fuels

Number of years left in global reserves

Coal

~100-150 years

Oil

~50 years

Natural Gas

~50 years

Describe the process of Fracking:

1.      Drill hole into shale rock

2.      Inject fracking fluid at high pressure to create cracks

3.      Extract newly released natural gas

4.      Seal fracking fluid in the hole

Nuclear Power

 

% of U-235

Uranium Ore

0.7%

Fuel-Grade

3.5%

Weapons-Grade

90%

Why is the concentration of U-235 important?

Only U-235 can undergo a fission chain reaction

What is done with the nuclear waste?

Stored on-site in spent fuel pools and/or concrete dry cask storage

 

Moderator

Control Rods

Slows down neutrons to be absorbed by U-235

Made from Water or Graphite (carbon)

Absorbs neutrons to limit number of chain reactions

Made from Boron

Renewable Energy

 

Variable Symbol

Unit

Power

P

W

Cross-Sectional Area

A

m2

Air Density

ρ

kg m-3

Air Speed

v

m s-1

Data Booklet Equations:

Photovoltaic Cells

Solar Concentrator

Solar Heating Panel

Converts solar energy directly into electricity. Useful in solar panels on top of building or solar farms connected to the energy grid

Mirrors focus sunlight onto a central tower. The high thermal energy is converted to steam and runs turbines to produce electricity

Sun’s radiation is absorbed by black pipes that transfer thermal energy to the water flowing through them. Replaces hot water heater.

 

Biomass

Coal

Geothermal

Hydropower

Natural Gas

Nuclear

Petroleum

Solar

Wind

Renewable

 

 

 

 

Produces CO2

 

 

 

 

 

 Thermal Energy Transfer

Conduction

Convection

Radiation

Energy is transferred through molecular collisions

Energy circulates through the expansion and rising of hot fluids

Energy is transferred through electromagnetic radiation. Can travel through a vacuum

 

Emissivity

 

Black Body Radiation

Sun

~1

 

An idealized object that absorbs all the electromagnetic radiation the falls on it

Earth

~0.6

 

Black-Body

1

 

Power Emissivity

Variable Symbol

Unit

Power

P

W

Emissivity

e

Surface Area

A

m2

Temperature

T

K

Max Wavelength

λmax

m

Data Booklet Equations:

Solar Radiation and Climate Change

Intensity

Variable Symbol

Unit

 

Data Booklet Equations:

Intensity

I

W m-2

 

Power

P

W

 

Area

A

m2

 

Greenhouse Gases

 

Positive Feedback Loop

Negative Feedback Loop

Water Vapor (H2O)

 

Melting ice (decreases albedo)

Cloud formation (increases albedo)

Carbon Dioxide (CO2)

 

Melting permafrost (releases methane)

Increased photosynthesis (uses CO2)

Methane (CH4)

 

Rising ocean temp releases methane

Climate Change leads to renewables

 

HEAT TRANSFER

Three modes of transmission of heat :
  • Conduction
  • Convection
  • Radiation

CONDUCTION

In thermal conduction particles of body at higher temperature transmit heat to the particles at lower temperature by mutual contact (collision) only and not by the movement of the particles. All solids are heated by conduction. Conduction cannot take place in vacuum.
Figure shows a solid of cross-section area A and thickness Δx. The face are at different temperature T1 and T2 (T1 > T2)
The rate of heat flow as found experimentally is given by
where
  …(1)
where K is proportionality constant called thermal conductivity.
It is a measure of how quickly heat energy can conduct through the substance.

 

Coefficient of thermal conductivity (K) : The coefficient of thermal conductivity, K, of a material is defined as the amount of heat that  flowing per second through a rod of that material having unit length and unit area of cross-section in the steady state, when the difference of temperature between two ends of the rod is 1 ºC and flow of heat is perpendicular to the end faces of the rod.  
Unit of coefficient of thermal conductivity in SI system is watt/m-K
Dimensions : [M L T–3 θ–1]

 

For a perfect conductor thermal conductivity K is infinite and for a perfect insulator K is zero.
In general, solids are better conductors than liquids and liquids are better conductors than gases. (Heat transfer through the process of conduction is possible in liquids and gases also, if they are heated from the top.) Metals are much better conductors than non-metals.
A good conductor of heat is also a good conductor of electricity. The conduction of both heat and electricity is due to the movement of free electrons.

CONVECTION

It is the process by which heat is transferred from one place to another in a medium by the movement of particles of the medium. It occurs due to density difference. This phenomenon occurs in fluids.

RADIATION

It is the process by which heat is transferred from one place to another without  any intervening medium. The light reaches from Sun to Earth by radiation process.
KEEP IN MEMORY
    1. The equation is valid for steady state condition. The condition is said to occur when no part of heat is used up in raising the temperature of any part of cross-section of the solid.
    2. On comparing equation (1) with the following equation used for flowing of charge on account of potential difference.
We find :
The role of resistance (thermal resistance) is played by
    1. Series combination of conductors
Equivalent thermal conductivity :
 
where H = heat flow per second
    1. Parallel combination of conductors
Equivalent thermal conductivity :
Keq =
    1. Davy’s safety lamp is based on the conduction. It is used in mines to know the ignition temperature of gases. Danger of explosion can be avoided.
    2. Principle of chimney used in a kitchen or a factory is based on the convection.
    3. Land and sea breezes are due to the convection.
    4. Temperature of the upper part of the flame is more than the temperature on the sides, because the currents of air carry the heat upwards.
    5. Radiation can be detected by differential air thermometer, Bolometer, thermopile, etc.

HEAT TRANSFER BY RADIATION AND NEWTON’S LAW OF COOLING

STEFAN’S LAW

This law is also called Stefan Boltzmann law. This law states that the power radiated for overall wavelength from a black body is proportional to the fourth power of thermodynamic temperature T.
for perfectly black body ….(1)
where E is energy emitted per second from unit surface area of the black body, T is temperature and σ is Stefan’s constant, and
σ = 5.67×10–8 Wm–2 K–4.
If body is not perfectly black body, then
E = eσT4 ….(2)
where e is emissivity. Then energy radiated per second by a body of area A is
E1 = eσT4 ….(3)
If the body temperature is T and surrounding temperature is T0, then net rate of loss of energy by body through radiation from equation (3) is
 ….(4)
Now rate of loss of energy in terms of specific heat c is
 ….(5)
and   ….(6)
(T- temperature, t – time in second)
So body cools by radiation and rate of cooling depends on e (emissivity of body), A (area of cross-section of body), m (mass of body) and c (specific heat capacity of body).
A black body is defined as a body which completely absorbs all the heat radiation falling on it without reflecting and transmitting any of it.

WIEN’S LAW

λmT = b,
where b is the Wien’s constant and b = 2.898 × 10–3mK
and T = temperature.

 

Graph of λm versus T
    
Hence on increasing temperature of a body, its colour changes gradually from red orange yellow green  blue violet. Thus the temperature of violet star is maximum and temperature of red star is minimum. Sun is a medium category star with λm = 4753Aº (yellow colour) and temperature 6000K.

KIRCHOFF’S LAW

According to this law, the ratio of emissive power to absorptive power is same for all surfaces at the same temperature and is equal to the emissive power of a perfectly black body at the same temperature.
i.e. 
where e = emissive power of a given surface
a = absorptive power of a given surface
E = emissive power of a perfect black body
A = absorptive power of a perfect black body
  • For a perfect black body, A = 1,
  • If emissive and absorptive power are considered for a particular wavelength λ then
  • Since Eλ is constant at a given temperature, according to this law if a surface is good absorber of a particular wavelength then it is also a good emitter of that wavelength.

FRAUNHOFER’S LINE

  • Fraunhofer lines are the dark lines in the spectrum of sun and are explained on the basis of Kirchoff’s law. When white light emitted from central core of sun (photosphere) passes through its atmosphere (chromosphere) radiation of those wavelength will be absorbed by the gases present there, which they usually emit (as good emitter is a good absorber) resulting dark lines in the spectrum of sun.
  • On the basis of Kirchoff’s law, Fraunhofer identified some of the elements present in the chromosphere. They are hydrogen, helium, sodium, iron, calcium, etc. Fraunhofer had observed about 600 darklines in the spectrum of sun.

NEWTON’S LAW OF COOLING

The rate of loss of heat of a body is directly proportional to the temperature difference between the body and the surroundings.
i.e.    
where T = temperature of body, T0 = temperature of surrounding.
and where, e = emissivity of body, A = area of surface of body, σ = Stefan’s constant, m = mass of body, c = specific heat of body.

 

It is a special case of Stefan’s law and above relation is applicable only when the temperature of body is not much different from the temperature of surroundings.

SOLAR CONSTANT

Solar constant is the solar radiation incident normally per second on one square metre at the mean distance of the earth from the sun in free space.
It is given by S = 1.34 × 109 Jm–2s–1.
Temperature of sun is given T4 = S/σ(R/r)2, where R is mean distance of the earth from the sun and r is the radius of the sun.
we know the temperature of the Sun we can use the Stefan–Boltzmann law to calculate the power radiated per unit area:

\(P= A\sigma T^4\)
\(\frac{P}{A} = 5.67\times 10^{-8} \times (5800)^4\)
\(=6.42\times 10^7\)  Watt
Radius of  \(Sun = 6.9 \times 10^8 \; m\)
Hence \( A = 4\pi r^2 = 6 \times 10^{18} \;m^2\)
Hence \( P = 3.9 \times 10^{26}\;W\)
Now distance of Earth from \(Sun = 1.5 \times 10^{11} \; m\)
so the power per unit area at the Earth will be:
\(I = \frac{P}{4\pi r^2}=\frac{3.9 \times 10^{26} }{4\pi(1.5 \times 10^{11})^2}\)
\(=1400 \; W\;m^{-2}\)

KEEP IN MEMORY
  • Good absorber is a good emitter.
  • Cooking utensils are provided with wooden or ebonite handles, since wood or ebonite is a bad conductor of heat.
  • Good conductor of heat are good conductors of electricity, Mica is an exception which being a good conductor of heat is a bad conductor of electricity.

The Atmosphere And Air Mass

The atmosphere scatters and absorbs some of the Sun’s energy that is incident on the Earth’s surface. Scattering of radiation by gaseous molecules (e.g. O2, O3, H2O and CO2), that are a lot smaller than the wavelengths of the radiation, is called Rayleigh scattering. Roughly half of the radiation that is scattered is lost to outer space, the remaining half is directed towards the Earth’s surface from all directions as diffuse radiation. Because of absorption by oxygen and ozone molecules the shortest wavelength that reaches the Earth’s surface is approximately 0.29 µm. Other gas molecules absorbed difference wavelengths as indicated in figure below.

Scattering by dust particles larger than wavelengths of light is called Mie scattering. This process includes both true scattering (where the radiation bounces of the particle) and absorption followed by emission, which heats the particles. The amount of radiation scattered by this process will vary a lot depending on location and the weather blowing particles about. A form of Mie scattering called the Tyndall effect, that preferentially scatters shorter wavelengths is responsible for the sky being blue.

Clouds reflect a lot of radiation and also absorbed a little, the rest is transmitted through. Globally, clouds reflect a lot of radiation and help regulate the surface temperature.

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