doPhysics · Thermal Physics 2

Conduction of Heat

Why this chapter

Why this chapter exists

Chai in a steel tumbler burns your fingers; the same chai in a clay kulhad does not. A quilt keeps you warm on a winter night. A lake in Ladakh freezes hard enough to walk on, but never solid to the bottom. All of these turn on one question: how fast does heat travel through a solid? This chapter answers it. It starts with what happens inside a solid when one end is hot. Then it finds the law of the heat current and turns it into a thermal resistance that adds like lengths of rod. With these tools it follows ice thickening on a lake and heat escaping through the lagging of a steam pipe.

You will learn

By the end of this chapter you will

Name the three ways heat travels, and explain conduction by atomic vibrations and by free electrons.
Describe the steady state of a rod and read its temperature gradient.
Use Fourier's law, \(H = -kA\,dT/dx\), and compare materials by their thermal conductivity.
Find thermal resistances and the heat currents and junction temperatures of rods in series, in parallel and in simple networks.
Derive and use the time for ice on a lake to grow from one thickness to another.
Find the heat current through the lagging of a pipe and through a spherical shell, and explain the shape of the temperature inside.

Before you begin

From the chapter on temperature you should know heat as energy in transit, and that water is densest at 4 °C. From calorimetry you should know latent heat, \(Q = mL\). From Mechanics you should know power as energy per second. Lessons 4 and 5 use simple integration: the integrals of \(x\), \(1/r\) and \(1/r^2\).

Already know this?

Lessons

How Heat Travels

Heat moves from hot to cold in three ways, by conduction, convection and radiation. This lesson looks at the first. In a solid, heat passes from atom to vibrating atom, and in a metal free electrons carry it much faster. That is why a steel tumbler of chai burns the fingers and a clay kulhad does not.

Three ways heat travelsConduction by molecules and electrons

12 min0%

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Fourier's Law

A rod between a hot body and a cold one warms until every slice passes on exactly the heat it takes in. Then its temperature falls in a straight line, and the heat current follows Fourier's law, \(H = kA\Delta T/L\). The thermal conductivity \(k\) sorts materials from copper to still air, and explains why a quilt keeps you warm.

Steady state and temperature gradientFouriers lawThermal conductivities

15 min0%

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Thermal Resistance and Rods in Series and in Parallel

A slab's thermal resistance, \(R = L/(kA)\), measures how many kelvin it takes to push one watt through it. Rods joined end to end share one heat current and their resistances add. Rods side by side share one temperature difference and their currents add. Symmetry settles a whole network of rods.

Thermal resistanceRods in seriesRods in parallelNetworks of rods

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The Growth of Ice on a Lake

A lake freezes from the top down, and the ice it has made slows its own growth. The latent heat of each new layer must be conducted up through the ice above it. This gives the time to thicken the ice, \(t = \rho L_f(x_2^2 - x_1^2)/(2k\theta)\), and shows why every centimetre takes longer than the one before.

Ice grows from the topTime to thicken the iceEach centimetre takes longer

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Heat Flow Through Pipes and Shells

When heat flows outward through the lagging of a pipe or the shell of a round vessel, the area it crosses grows with the radius. Fourier's law applied to thin shells gives \(H = 2\pi kL\Delta T/\ln(r_2/r_1)\) for a pipe and \(H = 4\pi kr_1r_2\Delta T/(r_2 - r_1)\) for a sphere. The temperature inside falls steeply near the inner surface and gently further out.

Radial flow through a cylinderRadial flow through a sphereTemperature inside the lagging

14 min0%

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Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.

The Workbook

Questions to solve on paper, marked by you.

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Glossary

The key words of this chapter, with the meaning this chapter gives them.

conduction: The flow of heat through matter from particle to particle, with no bulk motion of the matter itself. It is the way heat travels along the handle of a tawa.
convection: The flow of heat carried by the bulk motion of a fluid. The heated part expands, rises and is replaced by cooler fluid. It is treated in the next chapter.
equivalent conductivity: The conductivity of a single rod of the same total size that would carry the same heat current as a combination of rods. For two equal rods in series it is \(2k_1k_2/(k_1 + k_2)\).
free electrons: Electrons in a metal that are not bound to any one atom and move through the whole metal. They carry electric current and most of the heat a metal conducts.
heat current: The heat crossing a section per second, \(H = dQ/dt\), in watts.
junction temperature: The steady temperature at the joint of two rods in series. For rods of equal size between \(T_1\) and \(T_2\) it lies closer to the end of the better conductor.
lagging: A layer of a poor conductor, such as glass wool or foam. It is wrapped round a rod, pipe or vessel to stop heat escaping through its surface.
latent heat of fusion: The heat given out when one kilogram of a liquid freezes, or taken in when the solid melts, at a constant temperature. For water it is \(3.34 \times 10^5\,\text{J/kg}\).
parallel: An arrangement of rods or slabs side by side between the same two temperatures. Their heat currents add, and \(1/R = 1/R_1 + 1/R_2\).
radiation: The flow of heat as electromagnetic waves, which need no medium. It carries the Sun's heat across empty space. It is treated in the next chapter.
series: An arrangement of rods or slabs joined end to end, so the same heat current flows through each. Their thermal resistances add.
steady state: The state of a conducting body in which the temperature at every point no longer changes with time. Each part passes on all the heat it receives.
temperature gradient: The change of temperature per unit length along the direction of heat flow, \(dT/dx\), in K/m. It is negative when the temperature falls along \(x\).
thermal conductivity: The property of a material that fixes the heat current for a given area and temperature gradient, \(k\) in \(H = -kA\,dT/dx\). Copper has 400 W/(m K), still air 0.024 W/(m K).
thermal resistance: The temperature difference needed to drive one watt through a body, \(R = \Delta T/H\), in K/W. For a uniform lagged rod \(R = L/(kA)\).