Heat Transfer

Heat travels by three routes: conduction (usually in solids), convection (in liquids and gases) and radiation (needing no medium at all). First, conduction: why a metal spoon in hot tea soon burns your fingers.

How conduction works#

Put one end of a metal rod in a flame. Molecules at that end gain kinetic energy and, colliding with slower neighbours, share it. Those pass it on to their neighbours, and so on along the rod. Energy moves from molecule to molecule while each molecule stays, on average, in place: no matter flows.

Thermal conductivity#

A slab of cross-section A and thickness x has its faces held at T₁ and T₂, with its sides insulated. In the steady state, temperatures no longer change, and the same heat crosses every section per second: the heat current ΔQ/Δt. Experiment shows it is proportional to A and to T₁ − T₂, and inversely proportional to x:

For a varying cross-section or before steady state, apply it to a thin layer:

Solids generally conduct better than liquids, and liquids better than gases. Metals are far better still, because their free electrons carry energy rapidly through the metal.

Heat flow through a rod obeys the same arithmetic as electric current through a wire, so composite walls can be worked out like circuits.

Thermal resistance#

Write the heat current as i and the quantity x/KA as the thermal resistance R. Then equation 28.1 becomes

How to measure a metal’s conductivity in the lab, and the second route for heat: carried bodily by moving fluid.

Searle’s method for a good conductor#

A metal rod has one end in a steam chamber. Near the other end a copper tube is coiled round it, carrying a steady stream of water that enters at the far end and leaves nearer the steam. Two holes drilled in the rod, a distance x apart and filled with mercury, hold thermometers (θ₁, θ₂); two more measure the water out and in (θ₃, θ₄). Everything is lagged with felt or wool so no heat escapes from the sides.

Once all four readings are steady, collect the water flowing out for a time t (mass m) and measure the rod’s cross-section A. All the heat conducted along the rod ends up in the water:

Convection#

In convection the heated material itself moves and carries the heat with it. A hot-air blower heats air and a fan pushes it out. Water on a stove is heated at the bottom by conduction through the pan; it expands, becomes less dense, and rises, while cooler water sinks.

• Forced convection: the fluid is driven by a fan or pump.
• Natural (free) convection: the fluid moves because of density differences.

Ponds in winter

As the surface of a pond cools, the denser cold water sinks and warmer water rises to be cooled, until all the water reaches 4°C. Then surface water cooled further expands, stays on top and finally freezes. After that heat escapes only by conduction through the ice, a poor conductor, so freezing is slow and the bottom stays at 4°C: fish survive.

Inside the body

The body moves heat mainly by forced convection: the heart pumps blood around. It loses heat to the surroundings by conduction, convection and radiation, at a rate depending on clothing, sweating, air temperature, wind and humidity, yet keeps a remarkably steady temperature.

The sun warms us across 150 million kilometres of empty space. No conduction, no convection: radiation.

Thermal radiation#

Every body emits energy that travels through space like light and, when it falls on another body, is partly absorbed, raising that body’s thermal energy. This energy is called thermal radiation (or radiant energy). “Radiation” names both the process and the energy itself.

Prévost’s theory of exchange (1792)#

• All bodies radiate at all temperatures, at a rate depending on the surface, its area and its temperature (faster when hotter).
• They also absorb part of the radiation from surrounding bodies.
• Radiating more than it absorbs, a body cools; less, it warms.

A body left in a room settles at room temperature, yet is still radiating: it must now absorb exactly as fast as it emits. Put a hotter body in the room and it emits faster than it absorbs, so it cools; a colder one warms.

Blackbodies#

Hang a polished body and a black-painted one in a room. Both reach room temperature and receive the same radiation. The polished one reflects most of it and absorbs little; the black one absorbs most. Since neither changes temperature, the black one must also emit faster. Good absorbers are good emitters.

A perfect blackbody is an ideal; lampblack, reflecting about 1%, comes close. Better: a small hole in a cavity blackened inside, with a cone opposite the hole. Radiation entering the hole bounces round and is absorbed long before it can find its way out.

Kirchhoff’s law#

Proof: put a body A and an identical-shaped blackbody B in an enclosure, all at one temperature. Both receive radiation ΔU in time Δt. B absorbs all of it and, staying at constant temperature, emits the same: ΔU = kE₀. A absorbs aΔU and must emit as much: aΔU = kE. Dividing:

Heat a poker: first it only feels warm, then glows dull red, then orange, then white. Two laws describe this: one for the colour, one for the power.

What thermal radiation is#

Once emitted, thermal radiation is an electromagnetic wave like light. It obeys the laws of waves and, since its wavelength is small, travels in straight lines, casts shadows, reflects and refracts. A body emits a mixture of wavelengths, but most of the energy comes from a limited range.

At 300 K most of it is around 10 µm, deep in the infrared. Around 1100 K enough red is emitted for a body to glow red; around 3000 K enough shorter wavelengths for it to look white, though even then the peak is near 950 nm, still infrared.

Wien’s displacement law#

As the temperature rises, the wavelength of maximum intensity falls, in inverse proportion:

The Stefan–Boltzmann law#

The total energy radiated per second by a blackbody of area A at absolute temperature T is

Other bodies emit less, but still in proportion to T⁴:

Net loss to the surroundings#

In surroundings at T₀ the body absorbs eσAT₀⁴ (what it would emit at T₀), so its net loss is

A cup of tea cools quickly at first, then lingers lukewarm. Newton found the simple rule behind this, and two instruments let us detect radiation directly.

From radiation to Newton’s law#

The net radiation loss is eσA(T⁴ − T₀⁴). If T = T₀ + ΔT with ΔT small:

So radiation loss ≈ b₁A(T − T₀). Convection loss for small differences is also proportional to A(T − T₀). With mass m and specific heat s, the rate of fall of temperature is

Detecting thermal radiation#

The bolometer

Four identical grids cut from thin platinum foil form the arms of a Wheatstone bridge, balanced when no radiation falls. Grid 1 faces the radiation, arranged so whatever passes through its gaps lands on grid 4; grids 2 and 3 are shielded. Radiation heats grids 1 and 4, raising R₁ and R₄, so R₁R₄ > R₂R₃: the bridge unbalances and the galvanometer deflects, by an amount related to the radiation. Sealing it in an evacuated bulb makes it more sensitive.

The thermopile

Join two different metals at two junctions with a galvanometer in the circuit. If the junctions are at different temperatures, a current flows (the Seebeck effect): a thermocouple. A thermopile connects many antimony–bismuth thermocouples in series for sensitivity, with all the hot junctions on one blackened face exposed to the radiation (often behind a collecting cone) and the cold junctions shielded on the other.