Heat Transfer & Thermal Conduction

1. Newton's Law of Cooling

Sir Isaac Newton (1701) established that for small temperature differences between a body at temperature $T(t)$ and its ambient surroundings at constant temperature $T_s$, the rate of heat loss by combined convection and radiation is directly proportional to the temperature excess $(T - T_s)$:

where $h$ is the convective heat transfer coefficient (in $\text{W/(m}^2\cdot\text{K)}$) and $A$ is the exposed surface area.

Let the body have mass $m$ and specific heat capacity $c$, so $dQ = m c dT$:

where $k \equiv \frac{h A}{m c}$ is the cooling constant (in $\text{s}^{-1}$). Integrating from initial temperature $T_0$ at $t = 0$:

The temperature decays exponentially toward the ambient environment with relaxation time constant $\tau = 1/k$.

2. Physical Limits and Heat Capacity Models

Newton's cooling law is an approximation valid when $(T - T_s) \ll T_s$ (typically $\Delta T < 30^\circ\text{C}$). For large temperature differences, Stefan-Boltzmann radiation ($T^4 - T_s^4$) and non-linear turbulent natural convection introduce deviations.

Classical Dulong-Petit law states that at high temperatures, the molar heat capacity of all solid elements approaches $3R \approx 24.94\text{ J/(mol}\cdot\text{K)}$. At low temperatures, Einstein's single-frequency oscillator model and Debye's elastic continuum phonon model resolve the Third Law quantum freezing ($C_v \propto T^3$).

1. Fourier's Phenomenological Law of Conduction

Jean-Baptiste Joseph Fourier (1822) established that the conductive heat flux vector $\vec{q}$ (heat flow per unit area per unit time, in $\text{W/m}^2$) is linearly proportional to the negative local temperature gradient:

where $k$ is the thermal conductivity of the material (in $\text{W/(m}\cdot\text{K)}$). The minus sign ensures heat flows spontaneously down the temperature gradient from hot to cold in accordance with the Second Law.

2. Derivation of the 3D Heat Diffusion Equation

Consider an infinitesimal Cartesian volume element $dV = dx dy dz$ of density $\rho$ and specific heat $c_p$. By energy conservation:

The net conductive heat accumulation per unit volume is $-\nabla \cdot \vec{q} = \nabla \cdot (k \nabla T)$. If the material is isotropic and homogeneous ($k = \text{const}$), with internal volumetric heat source rate $\dot{q}$ (in $\text{W/m}^3$):

Dividing by $\rho c_p$:

where $\alpha \equiv \frac{k}{\rho c_p}$ is the Thermal Diffusivity (in $\text{m}^2\text{/s}$). Thermal diffusivity measures a material's capability to conduct thermal energy relative to its capacity to store thermal energy. In steady state without internal heat sources ($\partial T/\partial t = 0, \dot{q} = 0$), the heat equation reduces to Laplace's equation:

1. Rectilinear (1D Slab) Heat Conduction

For 1D steady flow along the $x$-axis through a flat slab of thickness $L$ and area $A$ with boundary conditions $T(0) = T_1$ and $T(L) = T_2$ ($T_1 > T_2$):

The total heat current $H = dq/dt$ is:

2. Radial Heat Flow Through a Coaxial Cylindrical Pipe

Consider heat conduction through the wall of a long hollow cylinder of inner radius $r_1$, outer radius $r_2$, length $L$, with temperatures $T_1$ and $T_2$:

The heat current crossing cylindrical area $A(r) = 2\pi r L$ is constant:

Integrating from $r_1$ to $r_2$:

Substituting $C_1$:

3. Radial Heat Flow Through Concentric Spherical Shells

For a hollow sphere of inner radius $r_1$ and outer radius $r_2$ with surface temperatures $T_1$ and $T_2$:

Heat current crossing spherical shell area $A(r) = 4\pi r^2$:

Integrating:

1. Ohm's Law Analogy for Thermal Conduction

Heat flow is mathematically analogous to electrical current flow:

• Temperature difference $\Delta T$ plays the role of electrical potential difference (voltage $V$).
• Heat current $H = dQ/dt$ plays the role of electric current $I$.
• Thermal Resistance $R_{\text{th}}$ plays the role of electrical resistance $R$:

For a flat slab of thickness $L$, area $A$, and conductivity $k$:

For a cylindrical shell: $R_{\text{th, cyl}} = \frac{\ln(r_2/r_1)}{2\pi k L}$. For a spherical shell: $R_{\text{th, sph}} = \frac{r_2 - r_1}{4\pi k r_1 r_2}$.

2. Series Multi-Layer Compound Walls

When heat flows sequentially through $n$ distinct material layers in series, the heat current $H$ is identical through every layer, and the total temperature drop is the sum of drops:

For a two-layer wall ($L_1, k_1$ and $L_2, k_2$), the interface junction temperature $T_j$ satisfies:

The equivalent thermal conductivity $k_{\text{eq}}$ for the total thickness $(L_1 + L_2)$ is:

1. Searle's Bar Method (Good Conductors)

For metals with high thermal conductivity (e.g., copper, aluminum), Searle's method uses a long cylindrical rod heated by steam at one end and cooled by circulating water at the other. Thermal insulation minimizes lateral losses.

• Temperatures $T_A$ and $T_B$ are measured at two points separated by distance $d$.
• Water enters the cooling jacket at $T_{\text{in}}$ and exits at $T_{\text{out}}$ with mass flow rate $\dot{m}$.

In steady state, heat conducted through the rod equals heat absorbed by cooling water:

2. Lee's Disc Method (Bad Conductors)

For thermal insulators (e.g., glass, cardboard, rubber), heat flow is small, making lateral losses significant. Lee's method places a thin disc of thickness $d$ and radius $r$ between a steam chest and a heavy brass base. After reaching steady state temperatures $T_1$ and $T_2$, the specimen is removed, the brass base is heated slightly above $T_2$, and its cooling curve $dT/dt$ is recorded:

3. The Wiedemann-Franz Law

In 1853, Gustav Wiedemann and Rudolf Franz discovered empirically that good electrical conductors are also good thermal conductors. In 1872, Ludvig Lorenz noted that the ratio of thermal conductivity $k$ to electrical conductivity $\sigma$ is directly proportional to absolute temperature $T$:

The constant $L_0$ is the Lorenz Number.

Arnold Sommerfeld (1927) derived $L_0$ from quantum Fermi-Dirac statistics for degenerate electron gases:

This universal constant holds for almost all metals at room temperature, demonstrating that heat and electrical charge in metals are transported by the exact same conduction electrons.