doPhysics · Thermal Physics 2
Radiation, Convection and Cooling
Heat that crosses empty space as radiation, and heat carried by moving fluids. Prevost's exchange and Kirchhoff's law, the Stefan-Boltzmann law and emissivity, the black-body spectrum and Wien's law, the temperatures of the Sun and the Earth, natural and forced convection, and Newton's law of cooling in its exact and average forms.
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Why this chapter
Why this chapter exists
The Sun warms your face across 150 million kilometres of empty space, where nothing can conduct heat. A clear desert night freezes while a cloudy one stays mild. A blacksmith reads the heat of iron from its colour, and a cup of chai cools fast at first and then ever more slowly. This chapter explains all of these. It starts with radiation: how much a body emits, at which wavelengths, and how much it takes back from its surroundings. With these laws it takes the temperature of the Sun and of the Earth. Then it turns to heat carried by moving air and water, and ends with the law that tells how any warm body cools.
You will learn
By the end of this chapter you will
- Explain thermal radiation and Prevost's exchange, and use \(a + r + t = 1\) and Kirchhoff's law, \(E = aE_b\).
- Use the Stefan-Boltzmann law, \(P = e\sigma AT^4\), and the net loss \(e\sigma A(T^4 - T_0^4)\).
- Read a black-body spectrum and use Wien's law, \(\lambda_{\max}T = b\), for glowing bodies and stars.
- Find the Sun's power and surface temperature from the solar constant, and the steady temperature of a planet.
- Explain natural and forced convection, the sea and land breezes, and use \(H = hA(T_s - T_f)\).
- State Newton's law of cooling, solve it for an exponential curve, and use the average-temperature form and its limits.
Before you begin
Before you begin
From the chapter on temperature you should know the kelvin scale and thermal equilibrium. From calorimetry you should know specific heat and heat capacity, \(Q = mc\Delta T\). From the chapter on conduction you should know the three ways heat travels. From Fluids you should know that a lighter fluid is pushed up by the upthrust. Lesson 6 uses a simple integral, of \(dx/x\).
Lessons
1
Thermal Radiation and Kirchhoff's Law
Every body radiates heat all the time, as waves that need no medium. Radiation falling on a surface is absorbed, reflected or passed on. Bodies never stop exchanging radiation, and a good absorber is always a good emitter, which makes a small hole in a box the blackest thing there is.
Thermal radiationAbsorptance reflectance transmittancePrevosts theoryKirchhoffs law and the black body
2
The Stefan-Boltzmann Law
A black body radiates in proportion to the fourth power of its absolute temperature, so doubling the temperature multiplies the power by sixteen. Real surfaces radiate a fraction e of that. What a body loses is the difference between what it emits and what it absorbs from its surroundings.
Stefan boltzmann lawEmissivity and grey bodiesNet radiation to surroundings
3
The Black-Body Spectrum and Wien's Law
A black body does not radiate all wavelengths equally. Its spectrum rises to a peak and falls, and as the body heats the whole curve grows and its peak slides to shorter wavelengths. That slide is why hot iron glows red, then orange, then white, and why a star's colour tells its temperature.
The black body spectrumWiens displacement lawColours and temperatures of stars
4
The Sun and the Earth
The sunlight reaching each square metre at the Earth, the solar constant, is enough to weigh the Sun's whole output and to take its temperature. The same law, balanced the other way, gives the temperature the Earth would have without its atmosphere, and shows what a rooftop solar panel can hope to collect.
The solar constantTemperature of the sunTemperature of the earth
5
Convection
Warm fluid expands, becomes lighter and rises, and cooler fluid takes its place. That circulation carries heat from a stove to a pan of dal, from the land to the sky over Chennai, and away from your skin under a ceiling fan. One coefficient measures how fast a surface hands its heat to a moving fluid.
Natural and forced convectionConvective heat transfer coefficientComparing the three ways
6
Newton's Law of Cooling
For a small excess over its surroundings, a body cools at a rate proportional to that excess, whether it loses heat by convection or by radiation. The excess then shrinks by the same factor in every equal interval, a cup of chai draws an exponential curve, and an average-temperature shortcut gives exam answers in two lines.
Newtons law of coolingThe exponential cooling curveThe average temperature methodComparing bodies by their cooling
Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.
0%The Workbook
Questions to solve on paper, marked by you.
Glossary
The key words of this chapter, with the meaning this chapter gives them.
- absorptance
- The fraction \(a\) of the radiation falling on a surface that the surface absorbs; also called its absorptive power. A black body has \(a = 1\).
- black body
- An ideal body that absorbs all the radiation falling on it, \(a = 1\). At any temperature it emits more radiation than any other body. A small hole in a hollow box is nearly one.
- black-body radiation
- The radiation given out by a black body, or by the hole of a heated cavity. Its amount and spectrum depend only on the temperature.
- black-body spectrum
- The graph of the power radiated per unit area per unit wavelength by a black body, against the wavelength. It has one peak, and its area is \(\sigma T^4\).
- convection
- The transfer of heat by the bulk motion of a fluid. The warmed fluid itself moves and carries its heat.
- convective heat transfer coefficient
- The constant \(h\) in \(H = hA(T_s - T_f)\), in W/(m² K). It is the heat carried per second from each square metre for each kelvin of difference; about 10 for still air.
- emissive power
- The energy radiated per second by each square metre of a surface, \(E\), in W/m². A black body at temperature \(T\) has \(E_b = \sigma T^4\).
- emissivity
- The ratio \(e = E/E_b\) of the emissive power of a surface to that of a black body at the same temperature. It lies between 0 and 1. By Kirchhoff's law it equals the absorptance.
- excess temperature
- The amount by which the temperature of a body exceeds that of its surroundings, \(T - T_0\). Newton's law of cooling makes the rate of cooling proportional to it.
- forced convection
- Convection in which a fan, a pump or the wind drives the fluid past the surface. It carries heat away much faster than natural convection.
- greenhouse effect
- The warming of a planet's surface by its atmosphere. Water vapour and carbon dioxide absorb the planet's infrared radiation and send part of it back down.
- grey body
- A body whose emissivity is the same at all wavelengths, so that it radiates \(P = e\sigma AT^4\).
- infrared
- Electromagnetic waves longer than red light, from about 0.7 μm to about 1 mm. Bodies at everyday temperatures radiate mainly in the infrared, which the eye cannot see.
- inverse-square law
- The rule that the power per unit area from a source falls as \(1/d^2\). The source's power spreads over a sphere of area \(4\pi d^2\).
- land breeze
- The night-time wind from the land to the sea. The land cools faster than the sea, the air over the sea rises, and air from the land flows out.
- luminosity
- The total power radiated by a star in all directions. For the Sun it is \(4\pi d^2S = 3.85 \times 10^{26}\,\text{W}\).
- natural convection
- Convection driven only by differences of density: warmed fluid is lighter, is pushed up by the upthrust, and cooler fluid takes its place.
- Newton's law of cooling
- For a small excess temperature, a body cools at a rate proportional to the excess. In symbols \(dT/dt = -k(T - T_0)\), with \(k = hA/(mc)\).
- reflectance
- The fraction r of the radiation falling on a surface that the surface reflects; also called its reflective power.
- sea breeze
- The daytime wind from the sea to the land. The land warms faster than the sea, the air over it rises, and cooler sea air flows in.
- solar constant
- The power of sunlight falling on each square metre facing the Sun at the Earth's average distance, above the atmosphere: about 1361 W/m².
- thermal equilibrium
- The state of bodies that have reached one common temperature. By radiation, each then emits as much as it absorbs, so the net flow is zero though the exchange continues.
- thermal radiation
- Energy radiated as electromagnetic waves by a body because of its temperature. It needs no medium and crosses a vacuum at the speed of light.
- transmittance
- The fraction \(t\) of the radiation falling on a body that passes through it; also called its transmissive power. With \(a\) and \(r\), \(a + r + t = 1\).
- Wien's displacement law
- The law \(\lambda_{\max}T = b\), with \(b = 2.9 \times 10^{-3}\,\text{m K}\). The wavelength at which a black body radiates most is inversely proportional to its absolute temperature.