Question

Long cylindrical AISI stainless steel rods $\left(k=7.74 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}^{\circ} \mathrm{F}\right.$ 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 $20 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}^{2}{ }^{\circ} \mathrm{F}$. If the rods enter the oven at \(70^{\circ} \mathrm{F}\), determine their centerline temperature when they leave. Solve this problem using the analytical one-term approximation method.

Short answer

Based on the given information, determine the Biot number (Bi), Fourier number (Fo), dimensionless temperature (θ), and the centerline temperature (T) of the steel rods when they leave the oven.

Step-by-step solution

Calculate the dimensionless parameters Biot number (Bi) and Fourier number (Fo)

(Note that all values provided are given in consistent units) First, we need to find the Biot number (Bi) and Fourier number (Fo), which are essential parameters in the analytical one-term approximation method. Biot number (Bi) can be calculated as: $$ Bi = \frac{h \cdot L_{c}}{k} $$ And Fourier number (Fo) can be calculated as: $$ Fo = \alpha \times \frac{t}{L_{c}^2} $$ Here, \(h\): Heat transfer coefficient given as \(20 \,\mathrm{Btu} /(\mathrm{h} \cdot \mathrm{ft}^{2} \cdot ^{\circ} \mathrm{F})\) \( k\): Thermal conductivity given as \(7.74 \,\mathrm{Btu} / (\mathrm{h} \cdot \mathrm{ft} \cdot ^{\circ} \mathrm{F})\) \(\alpha\): Thermal diffusivity given as \(0.135 \,\mathrm{ft^2} / \mathrm{h}\) \(L_c:\) Characteristic length, which is half of the diameter of the rod, \(L_c=0.5 \times D\) \(D\): Diameter of the rod, 4 inches; we must convert the diameter to ft, \(\frac{4 \, \text{in}}{12 \, \text{in/ft}} = \frac{1}{3} \, \text{ft}\) \(t\): residence time inside the oven \(v\): Velocity of the rod given as \(7 \,\mathrm{ft} / \mathrm{min}\), convert to \(\frac{7 \times 60}{1}\, \text{ ft/h}= 420 \,\mathrm{ft} / \mathrm{h}\) \(L_{oven}\): Length of the oven given as \(21\, \text{ft}\) We will first calculate the residence time of the rod in the oven: $$t = \frac{L_{oven}}{v}$$ Then we will calculate the Biot number (Bi) and Fourier number (Fo).

Calculate the dimensionless temperature θ*

The dimensionless temperature can be calculated using the analytical one-term approximation method, which is given by $$ θ = \frac{ T - T_{\infty} }{ T_i - T_{\infty} } = 2 \sum_{n = 1}^\infty (-1)^{n+1} \frac{1}{n} e^{-(n \pi)^2 \cdot Fo / 4 } \cdot \cos\left(\frac{(n \pi)}{2} \cdot Bi\right) $$ Here, we consider only n = 1 in the summation since higher terms will not considerably impact the final solution. \(θ\): Dimensionless temperature \(T\): Actual temperature \(T_i\): Initial temperature of the rods, given as \(70^{\circ} \mathrm{F}\) \(T_{\infty}\): Oven temperature given as \(1700^{\circ} \mathrm{F}\) After we calculate the dimensionless temperature, we can use the formula \(θ\) to find the actual temperature \(T\) at the centerline of the rod as it leaves the oven.

Calculate the final centerline temperature T

Using the formula for dimensionless temperature (θ), we can calculate the actual centerline temperature at the exit of the oven, \(T\), by rearranging the formula as follows: $$ T = θ \cdot (T_i - T_{\infty}) + T_{\infty}$$ We can now find the centerline temperature of the steel rods when they leave the oven using the calculated values from steps 1 and 2.