Question

Both Fourier's law of heat conduction and Fick's law of mass diffusion can be expressed as \(\dot{Q}=-k A(d T / d x)\). What do the quantities $\dot{Q}, k, A\(, and \)T\( represent in \)(a)\( heat conduction and \)(b)$ mass diffusion?

Short answer

(a) In heat conduction, Fourier's law is represented by the expression \(\dot{Q}=-k A(d T / d x)\). In this context, \(\dot{Q}\) represents heat flux (rate of heat transfer per unit area), \(k\) represents thermal conductivity (a material's ability to conduct heat), \(A\) represents the cross-sectional area through which heat transfer occurs, and \(T\) represents temperature distribution within the material. (b) In mass diffusion, Fick's law is given by the same expression \(\dot{Q}=-k A(d T / d x)\). Here, \(\dot{Q}\) stands for diffusion flux (rate at which particles diffuse per unit area), \(k\) represents the diffusion coefficient (a material's tendency of particles to diffuse), \(A\) denotes the cross-sectional area of particle diffusion, and \(T\) indicates the concentration distribution of particles within the material.

Step-by-step solution

Fourier's Law of Heat Conduction

For the case of Fourier's law of heat conduction, each variable represents the following: \(\dot{Q}\): Heat flux, which is the rate of heat transfer per unit area across a surface. \(k\): Thermal conductivity, a material property that indicates a substance's ability to conduct heat. \(A\): Cross-sectional area through which the heat is being transferred. \(T\): Temperature distribution within the material. The expression \(\dot{Q}=-k A(d T / d x)\) shows that the heat transfer rate is proportional to the temperature gradient within the material and the cross-sectional area of the material.

Fick's Law of Mass Diffusion

For the case of Fick's law of mass diffusion, each variable is as follows: \(\dot{Q}\): Diffusion flux, which represents the rate at which particles are diffusing through a unit area. \(k\): Diffusion coefficient (also known as diffusivity), a material property that quantifies the tendency of particles to diffuse. \(A\): Cross-sectional area through which the particles are diffusing. \(T\): Concentration distribution of the particles within the material. The expression \(\dot{Q}=-k A(d T / d x)\) demonstrates that the diffusion flux is proportional to the concentration gradient within the material and the cross-sectional area through which the particles are diffusing.