Long cylindrical AISI stainless steel rods \((k=\) \(7.74 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft} \cdot{ }^{\circ} \mathrm{F}\) and \(\left.\alpha=0.135 \mathrm{ft}^{2} / \mathrm{h}\right)\) of 4 -in-diameter are heat treated by drawing them at a velocity of \(7 \mathrm{ft} / \mathrm{min}\) through a 21 -ft-long oven maintained at \(1700^{\circ} \mathrm{F}\). The heat transfer coefficient in the oven is \(20 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}^{2} \cdot{ }^{\circ} \mathrm{F}\). If the rods enter the oven at \(70^{\circ} \mathrm{F}\), determine their centerline temperature when they leave. Solve this problem using analytical one-term approximation method (not the Heisler charts).

The Biot number (Bi) is a dimensionless parameter that indicates the relative significance of conduction within an object compared to convection on its surface. To put that in perspective, it's like comparing the influence of an object's internal thermal resistance against the resistance to heat transfer between the object's surface and the surrounding fluid.

In the context of the exercise, the Biot number is crucial in determining how to approach the problem of transient heat conduction. It's calculated using the formula: \[\text{Biot number } (Bi) = \frac{hL_{c}}{k}\]where \(h\) is the heat transfer coefficient, \(L_{c}\) is the characteristic length, and \(k\) is the thermal conductivity. If the Biot number is much less than 1, it indicates that the surface resistance to heat transfer dominates and the temperature within the object will be more uniform. However, a Biot number much larger than 1 suggests significant temperature gradients within the object.

In our solution, the characteristic length is taken to be the radius of the cylindrical rod since the heat transfer is radially symmetric. The Biot number in our calculation helped confirm that the one-term approximation method would be suitable for finding the centerline temperature.

The Fourier number (Fo), another crucial dimensionless number, represents the ratio of heat conduction rate to the heat storage rate within a material. Think of it as an indicator of how quickly the temperature within a material changes in comparison to the time it takes for heat to be conducted across the material. The Fourier number is calculated with the formula:\[\text{Fourier number } (Fo) = \frac{\alpha \tau}{L_{c}^{2}}\]Where \(\alpha\) is the thermal diffusivity, \(\tau\) is the time, and \(L_{c}\) is the characteristic length. In the exercise, we use the rod's radius as the characteristic length. The oven residence time plays a significant role here, indicating how long the rod is within the heat source, directly affecting the rod's temperature change over time.

The Fourier number is fundamental to transient heat conduction analysis because it enables the use of analytical solutions, like the one-term approximation method, to estimate the temperature at a specific time and location within the material, which is exactly what we did to determine the centerline temperature of the stainless steel rods.

The analytical one-term approximation method is a mathematical technique that simplifies the complex equations involved in transient heat conduction problems. Essentially, it forecasts the temperature distribution within an object based on time and spatial coordinates by using an approximate solution made up of a single term from an infinite series. This is particularly useful when exact solutions are either impossible or unnecessarily complex for practical applications.

To deploy the one-term approximation, we identify the initial and boundary conditions of the problem and then consult tables or charts with pre-calculated eigenvalues and coefficients pertinent to the geometry at hand—in this case, cylindrical. The method uses the following equation:\[\frac{T - T_{i}}{T_{\infty} - T_{i}} = \frac{4}{Bi} \sum_{n=1}^{\infty} \frac{C_{n} \sin(\lambda_{n}r_{*})\exp(-\lambda_{n}^{2} Fo)}{\lambda_{n}^{3}}\]In our exercise, we only needed the first term to estimate the rod's centerline temperature as it exits the oven. The one-term approximation is often sufficiently accurate for engineering purposes when the Biot and Fourier numbers are within certain ranges, which was rightly validated by our computation steps.