Question

Write down the relations for steady one-dimensional heat conduction and mass diffusion through a plane wall, and identify the quantities in the two equations that correspond to each other.

Short answer

Answer: The corresponding quantities in the two equations are: 1. Thermal conductivity (k) in heat conduction corresponds to the diffusion coefficient (D) in mass diffusion; 2. Temperature gradient (dT/dx) in heat conduction corresponds to the concentration gradient (dC/dx) in mass diffusion; 3. Heat transfer rate (q) in heat conduction corresponds to the mass transfer rate (J) in mass diffusion.

Step-by-step solution

(Step 1) Write the relation for steady one-dimensional heat conduction

For steady one-dimensional heat conduction through a plane wall, Fourier's law can be used to describe the heat transfer rate: q = -k * A * \frac{dT}{dx} where: q = heat transfer rate (W) k = thermal conductivity of the material (W/mK) A = cross-sectional area (m^2) \frac{dT}{dx} = temperature gradient (K/m)

(Step 2) Write the relation for steady one-dimensional mass diffusion

For steady one-dimensional mass diffusion through a plane wall, Fick's first law can be used to describe the mass transfer rate: J = -D * A * \frac{dC}{dx} where: J = mass transfer rate (kg/s) D = diffusion coefficient (m^2/s) A = cross-sectional area (m^2) \frac{dC}{dx} = concentration gradient (kg/m^3 per m)

(Step 3) Identify the corresponding quantities in the two equations

Now, let's compare Fourier's law and Fick's first law to identify the corresponding quantities in the two relations. In both relations: - The negative sign indicates that the heat or mass transfer occurs in the direction of decreasing temperature or concentration gradient; - A is the cross-sectional area through which heat or mass transfer occurs; - \frac{dT}{dx} and \frac{dC}{dx} are the driving forces for heat and mass transfer, respectively; Therefore, the corresponding quantities in the two equations are: - k in heat conduction ↔ D in mass diffusion: they represent the transport property of the material (thermal conductivity and diffusion coefficient, respectively); - \frac{dT}{dx} in heat conduction ↔ \frac{dC}{dx} in mass diffusion: they represent the gradient responsible for the transport (temperature gradient and concentration gradient, respectively); - q in heat conduction ↔ J in mass diffusion: they represent the transport rate (heat transfer rate and mass transfer rate, respectively).