Question

Write down the one-dimensional transient heat conduction equation for a long cylinder with constant thermal conductivity and heat generation, and indicate what each variable represents.

Short answer

Question: Write the one-dimensional transient heat conduction equation for a long cylinder with constant thermal conductivity and heat generation, and indicate the meaning of each variable in the equation. Answer: The one-dimensional transient heat conduction equation for a long cylinder with constant thermal conductivity and heat generation is given by: ρC_p(∂T/∂t) = (k/r)(∂/∂r)(r∂T/∂r) + Q where: - T: Temperature (K) - t: Time (s) - r: Radial position from the center of the cylinder (m) - ρ: Density (kg/m^3) - C_p: Specific heat at constant pressure (J/kgK) - k: Thermal conductivity (W/mK) - Q: Heat generation per unit volume (W/m^3)

Step-by-step solution

Write the general equation for transient heat conduction

The general equation for transient heat conduction is given by Fourier's law: q = -k∇T where, - q is the heat flux (W/m^2) - k is the thermal conductivity (W/mK) - T is the temperature (K) - ∇ is the gradient operator The unsteady-state heat conduction equation in the Cartesian coordinate system can be written as: ρC_p(∂T/∂t) = k(∂^2T/∂x^2 + ∂^2T/∂y^2 + ∂^2T/∂z^2) + Q where, - ρ is the density (kg/m^3) - C_p is the specific heat at constant pressure (J/kgK) - Q is the heat generation per unit volume (W/m^3)

Modify the equation for a cylindrical coordinate system

Now we will rewrite the equation in cylindrical coordinates for a long cylinder, considering heat transfer only in the radial direction (r): ρC_p(∂T/∂t) = k(1/r ∂/∂r(r∂T/∂r)) + Q

Take into account constant thermal conductivity and heat generation

Since the thermal conductivity is constant, we can simplify the equation: ρC_p(∂T/∂t) = (k/r)(∂/∂r)(r∂T/∂r) + Q This is the one-dimensional transient heat conduction equation for a long cylinder with constant thermal conductivity and heat generation.

Describe the meaning of each variable

In the final equation: - T: Temperature (K) - t: Time (s) - r: Radial position from the center of the cylinder (m) - ρ: Density (kg/m^3) - C_p: Specific heat at constant pressure (J/kgK) - k: Thermal conductivity (W/mK) - Q: Heat generation per unit volume (W/m^3)