Question

The temperature of the cylinder (with radius r, length -h���z���h) is constant at to. The cylinder is suddenly immersed into a fluid. The temperature of the fluid is constant at t[infinity], and the convective coefficient of the fluid is h. The density, heat capacity, and conductivity of the cylinder are p, cp, and k, respectively. Assume the properties of the cylinder are constant. Derive the partial differential equation of the temperature of the solid (from shell balance), and write down initial and boundary conditions.

Answer 1

To derive the partial differential equation for the temperature of the cylinder, a shell balance approach leads to the heat diffusion equation in cylindrical coordinates. Initial condition is the constant initial temperature throughout the cylinder, while boundary conditions incorporate symmetry at the center and convective heat transfer at the surface.

Step-by-step explanation:

Deriving the Partial Differential Equation for Heat Conduction

To derive the partial differential equation for the temperature of the cylinder, we use an energy balance that equates the rate of heat energy entering a differential volume element with the rate of heat energy stored plus the rate of heat energy leaving.

Considering a cylindrical shell at a radius r with thickness dr within the cylinder, conducting heat radially without any generation or consumption of heat, the equation is found by Fourier's law of heat conduction.

Steps to derive the PDE:

Establish the thermal energy entering through the inner face of the shell at radius r, Q_in, using Fourier's law.

Establish the thermal energy leaving through the outer face of the shell at radius r + dr, Q_out.

Write down the conservation of energy principle, stating that the rate of heat accumulation must equal the rate of heat entering minus the rate of heat leaving.

Apply the assumption of constant properties (p, cp, k) within the cylinder to simplify the equations.

Use the definition of the heat transfer rate to express the conservation of energy in the form of a differential equation.

End up with the partial differential equation, identifying the heat diffusion equation in cylindrical coordinates.

Initial and Boundary Conditions:

The initial condition is:

• The initial temperature throughout the cylinder is constant at t_o.

The boundary conditions are:

At the centerline of the cylinder, symmetry dictates that the radial derivative of temperature with respect to r is zero.

At the surface of the cylinder (r = radius), the convective heat transfer boundary condition applies, relating the surface temperature to the temperature of the surrounding fluid and the convective heat transfer coefficient h.

The temperature at infinity is constant at t_infinity.