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.

For a long cylinder, we will use the cylindrical coordinate system. The one-dimensional heat conduction equation, considering internal heat generation and constant thermal conductivity, is given by: \(\frac{1}{r}\frac{\partial}{\partial r}(r \frac{\partial T}{\partial r}) + \frac{q_g}{k} = \rho C_p \frac{\partial T}{\partial t}\) where: T: Temperature (of the cylinder) r: Radial distance from the center of the cylinder t: Time k: Thermal conductivity of the material (constant) \(q_g\): Heat generation per unit volume \(\rho\): Density of the material \(C_p\): Specific heat capacity of the material

Each variable in the heat conduction equation represents the following: - T (Temperature): It describes the temperature distribution within the cylinder as a function of radial distance and time. - r (Radial distance): It represents the distance from the center of the cylinder. We are only considering the one-dimensional heat transfer in the radial direction. - t (Time): It represents the time variable and helps us to understand how the temperature distribution is changing with time. - k (Thermal conductivity): It is a material property that determines how efficiently heat is conducted through the material. For a constant thermal conductivity, this value does not change with temperature. - \(q_g\) (Heat generation per unit volume): It represents the rate at which heat is generated within the material per unit volume. For a constant heat generation, this value does not change with temperature, distance, or time. - ρ (Density): It is a material property that represents the mass per unit volume of the material. - \(C_p\) (Specific heat capacity): It is a material property that measures the amount of heat required to raise the temperature of a unit mass of the material by one degree Celsius/Kelvin.