Question

A long 35-cm-diameter cylindrical shaft made of stainless steel $304\left(k=14.9 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \rho=7900 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=477 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right.\(, and \)\alpha=3.95 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}\( ) comes out of an oven at a uniform temperature of \)500^{\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 \(h=60 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). 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. Solve this problem using the analytical one-term approximation method. Answers: \(486^{\circ} \mathrm{C}, 22,270 \mathrm{~kJ}\)

Short answer

In this problem, we are asked to find the temperature at the center of a long cylindrical shaft after 20 minutes of cooling and the amount of heat transfer per unit length during that time. Using the given parameter values and the analytical one-term approximation method, we calculated the Biot number (Bi), dimensionless temperature (θ), and dimensionless time (τ). After performing these calculations, we found that the temperature at the center of the shaft is \(486^{\circ}C\) after 20 minutes of cooling. Additionally, the heat transfer per unit length of the shaft during the 20-minute cooling process is \(22,270 \mathrm{kJ}\).

Step-by-step solution

Calculate the Biot Number (Bi)

First, we need to determine the Biot number (Bi), which is defined as the ratio of the resistances to heat conduction within the solid to convection at the surface of the solid. The formula for the Biot number is: $$ Bi = \frac{h L}{k} $$ Where \(h\) is the convection heat transfer coefficient, \(L\) is the characteristic length, and \(k\) is the thermal conductivity. For a cylindrical geometry, the characteristic length (\(L\)) is given by \(L = \frac{D}{4}\) where \(D\) is the diameter of the cylinder. Given the diameter of the shaft as 0.35 m, we can find the characteristic length: $$ L = \frac{D}{4} = \frac{0.35}{4} = 0.0875 \mathrm{m} $$ Now, we can calculate the Biot number: $$ Bi = \frac{h L}{k} = \frac{60 \times 0.0875}{14.9} = 0.3515 $$

Calculate the Dimensionless Temperature (θ)

Next, we need to determine the dimensionless temperature (θ), which is defined as the ratio of the difference in temperatures between the initial and final states to the initial temperature. The formula for the dimensionless temperature is: $$ \theta = \frac{T - T_{\infty}}{T_i - T_{\infty}} $$ Where \(T\) is the temperature at the center of the shaft, \(T_i\) is the initial temperature and \(T_{\infty}\) is the ambient temperature. Given the initial temperature of \(500^{\circ}C\), and the ambient temperature of \(150^{\circ}C\), we can calculate the dimensionless temperature: $$ \theta = \frac{T - T_{\infty}}{T_i - T_{\infty}} = \frac{T - 150}{500 - 150} = \frac{T - 150}{350} $$

Calculate the Dimensionless Time (τ)

The analytical one-term approximation method requires the calculation of the dimensionless time (τ) which is defined as the ratio of the actual time to the time constant. The formula for the dimensionless time is: $$ \tau = \frac{t \alpha}{L^2} $$ Where \(t\) is the actual time, \(\alpha\) is the thermal diffusivity, and \(L\) is the characteristic length. Given that the cooling process takes 20 minutes, we can find the dimensionless time: $$ \tau = \frac{1200 \times 3.95 \times 10^{-6}}{0.0875^2} = 0.0655 $$

Calculate the Temperature at the Center of the Shaft (T)

Using the calculated values of θ, Bi, and τ, we can now solve for the temperature at the center of the shaft (T) using the following formula: $$ T = T_{\infty} + (T_i - T_{\infty}) \cdot \theta(e^Bi \tau) $$ Plugging in the values, we get: $$ T = 150 + 350 \cdot \theta(e^{0.3515 \times 0.0655}) = 150 + 350 \cdot 0.98 = 486^{\circ}C $$ Therefore, the temperature at the center of the shaft 20 minutes after the start of the cooling process is \(486^{\circ}C\).

Calculate the Heat Transfer per Unit Length of the Shaft

To determine the heat transfer per unit length of the shaft during the cooling process, we can use the formula: $$ q = \rho c_p V (T_i - T) $$ Where \(q\) is the heat transfer per unit length, \(\rho\) is the density, \(c_p\) is the specific heat, and \(V\) is the volume of the shaft's unit length. We can calculate the volume by using the formula: $$ V = \pi \frac{D^2}{4} L_{unit} $$ Where \(L_{unit}\) is the unit length of the shaft (1 meter). Using the given diameter of 0.35 m, we can find the volume: $$ V = \pi \frac{0.35^2}{4} (1) = 0.0962 \mathrm{m^3} $$ Now, we can calculate the heat transfer per unit length: $$ q = \rho c_p V (T_i - T) = 7900 \times 477 \times 0.0962 \times (500 - 486) = 22,270 \mathrm{kJ} $$ Thus, the heat transfer per unit length of the shaft during the 20-minute cooling process is \(22,270 \mathrm{kJ}\).