A long 35-cm-diameter cylindrical shaft made of stainless...

A long 35-cm-diameter cylindrical shaft made of stainless steel $304\left(k=14.9 \mathrm{~W} / \mathrm{m} \cdot{ }^{\circ} \mathrm{C}, \rho=7900 \mathrm{~kg} / \mathrm{m}^3\right.$, $c_p=477 \mathrm{~J} / \mathrm{kg} \cdot{ }^{\circ} \mathrm{C}$, and $\left.\alpha=3.95 \times 10^{-6} \mathrm{~m}^2 / \mathrm{s}\right)$ comes out of an oven at a uniform temperature of $400^{\circ} \mathrm{C}$. The shaft is then allowed to cool slowly in a chamber at $150^{\circ} \mathrm{C}$ with an average convection heat transfer coefficient of $h=60 \mathrm{~W} / \mathrm{m}^2 \cdot{ }^{\circ} \mathrm{C}$. Determine the temperature at the center of the shaft 20 min after the start of the cooling process. Also, determine the heat transfer per unit length of the shaft during this time period.

Key Concepts

Eigenfunction Expansion in Cylindrical Coordinates

When solving transient conduction problems in cylindrical geometries, eigenfunction expansions, often involving Bessel functions, are employed. These solutions account for the radial temperature distribution and allow for the determination of the center temperature as a function of time.

Fourier Number

The Fourier number is a dimensionless time parameter used in heat conduction analysis. It represents the ratio of heat conduction rate to the rate of thermal energy storage, helping to characterize the transient thermal response of the material over time.

Biot Number

The Biot number is a dimensionless parameter that compares the internal conductive resistance of a body to the external convective resistance at its surface. It helps in determining whether spatial temperature gradients inside the body are significant, thereby indicating if a lumped capacitance approach is appropriate or if a full spatial analysis is required.

Heat Transfer per Unit Length Calculation

This involves applying an energy balance to determine the total amount of heat lost per unit length of the cylinder. It relates the integrated temperature change over the volume of the shaft to the convective heat flux at the surface, thereby quantifying the energy transfer during the cooling process.

Convection Heat Transfer

Convection is the mechanism by which heat is transferred from a solid surface to a surrounding fluid. A convective boundary condition, characterized by a heat transfer coefficient, is imposed at the surface of the shaft, linking the surface temperature to the ambient temperature and governing the rate at which the shaft loses heat.

Transient Heat Conduction

This concept involves analyzing the time?dependent variation of temperature within a solid. It is governed by the heat diffusion (conduction) equation, which, in cylindrical geometries, must account for spatial variations in temperature and is typically solved using methods like separation of variables, resulting in series solutions involving eigenfunctions.

Thermal Properties of Materials

Properties such as thermal conductivity, density, specific heat, and thermal diffusivity are fundamental in determining how a material conducts and stores heat. They influence the rate of temperature change within the solid and are crucial inputs in solving transient heat conduction problems.

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