Question

A hot dog can be considered to be a \(12-\mathrm{cm}\)-long cylinder whose diameter is \(2 \mathrm{~cm}\) and whose properties are $\rho=980 \mathrm{~kg} / \mathrm{m}^{3}\(, \)c_{p}=3.9 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}, k=0.76 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\(, and \)\alpha=2 \times 10^{-7} \mathrm{~m}^{2} / \mathrm{s}\(. A hot dog initially at \)5^{\circ} \mathrm{C}\( is dropped into boiling water at \)100^{\circ} \mathrm{C}$. The heat transfer coefficient at the surface of the hot dog is estimated to be \(600 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). If the hot \(\operatorname{dog}\) is considered cooked when its center temperature reaches \(80^{\circ} \mathrm{C}\), determine how long it will take to cook it in the boiling water. Solve this problem using the analytical one-term approximation method.

Short answer

Short Answer: It will take approximately 6 minutes and 9 seconds to cook the hot dog in boiling water until its center temperature reaches 80°C.

Step-by-step solution

Calculate the initial and final temperature differences

First, we calculate the initial and final temperature differences using the given values: \(\Theta_{i} = T_{i} - T_{\infty} = 5 - 100 = -95^{\circ} \mathrm{C}\) \(\Theta_{\infty} = T_{c} - T_{\infty} = 80 - 100 = -20^{\circ} \mathrm{C}\)

Calculate the perimeter P and cross-sectional area A

Next, we determine the cross-sectional area \(A\) and perimeter \(P\): \(A = \frac{\pi d^2}{4} = \frac{\pi (0.02)^2}{4} = 3.142 \times 10^{-4} \mathrm{~m}^2\) \(P = \pi d = \pi (0.02) = 0.0628 \mathrm{~m}\)

Calculate the constant C1

Then, we calculate the constant \(C_1\) by using the heat transfer coefficient \(h\), thermal conductivity \(k\), perimeter \(P\), and cross-sectional area \(A\): \(C_1 = \frac{2 \sqrt{h P k A}}{k A} = \frac{2 \sqrt{(600)(0.0628)(0.76)(3.142 \times 10^{-4})}}{(0.76)(3.142 \times 10^{-4})} = 37.8\)

Find the first root of the Bessel function (b1)

We can find the first root of the Bessel function of first kind using a table or online calculator. Here, we'll use the value: \(b_1 ≈ 2.4048\)

Compute the time to cook t

Finally, we substitute the values we found into the formula to find the time required to cook the hot dog: \(t = \frac{L^2}{12\alpha}\left(\frac{b_1^2\pi^2}{\ln\left(\frac{1+\Theta_{i} C_1}{1+\Theta_{\infty} C_1}\right)}-1\right) = \frac{(0.12)^2}{12(2 \times 10^{-7})}\left(\frac{(2.4048)^2\pi^2}{\ln\left(\frac{1+(-95)(37.8)}{1+(-20)(37.8)}\right)}-1\right) ≈ 368.87 \mathrm{~s}\) Therefore, it will take about 368.87 seconds, or approximately 6 minutes and 9 seconds, to cook the hot dog in the boiling water.