Rates of Processes

The study of nonequilibrium processes, traditionally called transport theory or kinetics, is still a field of active research. Here the focus is on heat transfer. Radiation, which crosses even empty space, is left for later, and convection is too complicated for now. That leaves conduction: at the molecular level, fast molecules in a gas or liquid collide with slow ones and pass on energy, while in a solid the energy is carried by vibrations of the lattice and, in metals, by conduction electrons.

The phenomenological theory is simple. The heat crossing a slab of area \(A\) and thickness \(\Delta x\) in time \(\Delta t\) is proportional to \(A\,\Delta T\,\Delta t/\Delta x\); the constant of proportionality is the thermal conductivity \(k_t\):

This is the Fourier heat conduction law (the same Fourier as in Fourier analysis). The minus sign says heat flows downhill: if \(T\) increases from left to right, heat flows from right to left.

Problem 1.62. In a uniform rod whose temperature varies only along its length, the net heat flowing into a short segment \(\Delta x\) is the difference between what enters from one side and leaves from the other, \(Q_{\text{net}} = k_tA\,(\partial^2T/\partial x^2)\,\Delta x\,\Delta t\). It raises the segment’s temperature by \(Q_{\text{net}}/C\), with heat capacity \(C = c\rho A\,\Delta x\) (specific heat \(c\), density \(\rho\)), giving

the heat equation, a general form of the diffusion equation. It has the solution

a pulse of heat that spreads out and flattens, settling toward the background temperature \(T_0\) as \(t \to \infty\). Below, the rod is solved numerically; its ends are insulated, so no energy leaves. Click or tap on the rod to add heat anywhere.

How far does a gas molecule travel before it hits another? Treat the other molecules as points and give the moving one radius \(2r\): it sweeps out a cylinder of volume \(\pi(2r)^2\ell\), and when that equals the average volume per molecule, a collision is likely. That defines the mean free path:

For nitrogen or oxygen, \(r \approx 1.5 \times 10^{-10}\) m; at room temperature and pressure \(V/N = k_{\mathrm{B}}T/P \approx 4 \times 10^{-26}\ \mathrm{m^3}\), so \(\ell \approx 150\ \mathrm{nm}\), about 40 times the average spacing between molecules, and the average time between collisions is \(\overline{\Delta t} = \ell/\bar{v} \approx 3 \times 10^{-10}\ \mathrm{s}\).

Imagine a line with box 1 (energy \(U_1\)) on one side and box 2 (\(U_2\)) on the other, each one mean free path wide. In a time \(\Delta t\), about half the molecules in each box cross the line, in random directions, so the net heat flow is

Comparing with Fourier’s law, and using \(\bar{v} = \ell/\Delta t\),

Since \(\ell \propto V/N \propto T/P\), the pressure and the explicit \(T\) cancel, leaving \(k_t \propto \bar{v} \propto \sqrt{T}\), independent of pressure.

Slide one plate over another with fluid between them, slowly and with a narrow gap, so the flow is laminar (entirely horizontal). The top layer of fluid gives up some of its forward momentum to the layer below, and so on down, so a backward force acts on the top plate (and, by Newton’s third law, a forward one on the bottom). The simplest guess, \(F_x/A \propto \Delta u_x/\Delta z\), gives in differential form

where \(u_x\) is the macroscopic flow velocity (not the much faster thermal velocities) and \(\eta\) is the viscosity. The left side has units of pressure (Pa) but is not a pressure: it acts parallel to the surface, a shear stress.

Viscosity is momentum transport, just as conduction is energy transport. In a gas, as for \(k_t\), the momentum density (\(\propto N/V\)) and the mean free path (\(\propto V/N\)) cancel, so \(\eta \propto \sqrt{T}\): gases get more viscous as they warm, because their molecules move faster. Liquids do the opposite. At low temperature their molecules latch onto each other better as they collide, transferring momentum very efficiently; the extreme case is a solid, whose molecules are bonded together permanently.

Heat conduction transports energy and viscosity transports momentum; diffusion transports particles, which tend to spread from where their concentration \(n\) (particles per unit volume) is high to where it is low. In analogy with the other two, the flux \(\vec{J}\), the net number of particles crossing a surface per unit area per unit time, obeys

Fick’s law, with a diffusion coefficient \(D\) that depends on both the molecules diffusing and what they diffuse through. Considering the flux into and out of a short segment of a pipe gives Fick’s second law (Problem 1.69), the same equation as for heat:

Below, each dye particle takes small random steps, independently of all the others. Nothing pushes them from high to low concentration, yet the histogram follows the smooth solution of Fick’s second law.

Roughly, for \(N\) particles crossing a region of size \(\Delta x\), \(N/(A\,\Delta t) \approx D\,(N/V)/\Delta x\) with \(V \approx A\,\Delta x\), so \(\Delta t \approx \Delta x^2/D\). The square makes diffusion fast over short distances and hopelessly slow over long ones. For dye in water, \(D \approx 10^{-9}\ \mathrm{m^2/s}\), and spreading over 10 cm takes \(10^7\) s, about 4 months. Ink in a glass mixes in minutes because of the bulk motion of the water (convection), not diffusion.