In Betty Crocker's Cookbook, it is stated that it takes \(5 \mathrm{~h}\) to roast a \(14-\mathrm{lb}\) stuffed turkey initially at \(40^{\circ} \mathrm{F}\) in an oven maintained at \(325^{\circ} \mathrm{F}\). It is recommended that a meat thermometer be used to monitor the cooking, and the turkey is considered done when the thermometer inserted deep into the thickest part of the breast or thigh without touching the bone registers \(185^{\circ} \mathrm{F}\). The turkey can be treated as a homogeneous spherical object with the properties \(\rho=75 \mathrm{lbm} / \mathrm{ft}^{3}, c_{p}=0.98 \mathrm{Btu} / \mathrm{lbm} \cdot{ }^{\circ} \mathrm{F}\), \(k=0.26 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft} \cdot{ }^{\circ} \mathrm{F}\), and \(\alpha=0.0035 \mathrm{ft}^{2} / \mathrm{h}\). Assuming the tip of the thermometer is at one- third radial distance from the center of the turkey, determine \((a)\) the average heat transfer coefficient at the surface of the turkey, \((b)\) the temperature of the skin of the turkey when it is done, and \((c)\) the total amount of heat transferred to the turkey in the oven. Will the reading of the thermometer be more or less than \(185^{\circ} \mathrm{F} 5\) min after the turkey is taken out of the oven?

We are given that the thermometer is inserted at one-third of the radial distance, which means \(\xi = r / R = 1/3\). The given time for roasting the turkey is \(5 \mathrm{~h}\), so \(\tau = \frac{\alpha t}{r^2} = \frac{0.0035 \times 5}{(1/3)^2} = 15.75\).

Using the given equation: $$T(\xi, \tau) = (T_{\text{surface}}-325) \left[ 1-\frac{2}{\sqrt{\pi}} \sum_{n=1}^{\infty}(-1)^n \frac{e^{-n^2 \xi}}{n} \right] + 325$$ Setting \(T(\xi, \tau) = 185\), we have: $$(185-325) \left[ 1-\frac{2}{\sqrt{\pi}} \sum_{n=1}^{\infty}(-1)^n \frac{e^{-n^2 \xi}}{n} \right] + 325 = 0$$ Solving for \(T_{\text{surface}}\), we get \(T_{\text{surface}} \approx 414.6^{\circ} \mathrm{F}\). Now, we can find the average heat transfer coefficient, \(h\), using the formula: $$h = \frac{4}{3} \frac{k}{R}$$ First, we need to find the turkey's radius, \(R\). Given the mass (\(m = 14\mathrm{~lb}\)) and density (\(\rho = 75\mathrm{~lbm}/\mathrm{ft}^3\)), we have: $$R = \left(\frac{3m}{4\pi\rho}\right)^{1/3} = \left(\frac{3 \times 14}{4\pi \times 75}\right)^{1/3} \approx 0.376\mathrm{~ft}$$ Now, calculate the average heat transfer coefficient, \(h\): $$h = \frac{4}{3} \frac{k}{R} = \frac{4}{3} \frac{0.26}{0.376} \approx 0.918\mathrm{~Btu}/\mathrm{h} \cdot \mathrm{ft}^2 \cdot{ }^{\circ} \mathrm{F}$$ So, the average heat transfer coefficient, \(h \approx 0.918\mathrm{~Btu}/\mathrm{h} \cdot \mathrm{ft}^2 \cdot{ }^{\circ} \mathrm{F}\).

The total amount of heat transferred to the turkey, \(Q\), can be calculated as: $$Q = mc_p(T_{\text{final}} - T_{\text{initial}})$$ Given that the turkey is cooked well at \(185^{\circ} \mathrm{F}\), we can calculate the total amount of heat transferred as: $$Q = 14 \times 0.98 \times (185-40) \approx 1971.08\mathrm{~Btu}$$ So, the total amount of heat transferred to the turkey during cooking is about \(1971.08\mathrm{~Btu}\).

The turkey is taken out of the oven, and the temperature of the thermometer is expected to decrease with time. We need to find the temperature 5 minutes (\(t = 5/60\mathrm{~h}\)) after the turkey is removed from the oven. We can use the same equation, with the change in time \(\tau\) and the oven temperature \(T_{\infty} = 72^{\circ} \mathrm{F}.\)Calculate the new dimensionless time \(\tau_{\text{after}} = \frac{\alpha t}{r^2} = \frac{0.0035 \times 5/60}{(1/3)^2} = 0.525\). Recompute the temperature: $$T(\xi, \tau_{\text{after}}) = (T_{\text{surface}}-72) \left[ 1-\frac{2}{\sqrt{\pi}} \sum_{n=1}^{\infty}(-1)^n \frac{e^{-n^2 \xi}}{n} \right] + 72$$ Calculating the temperature, we get \(T(\xi, \tau_{\text{after}}) \approx 183.5^{\circ} \mathrm{F}\), which is less than \(185^{\circ} \mathrm{F}\). In conclusion: \((a)\) The average heat transfer coefficient at the surface of the turkey is approximately \(0.918\mathrm{~Btu}/\mathrm{h} \cdot \mathrm{ft}^2 \cdot{ }^{\circ} \mathrm{F}\). \((b)\) The temperature of the skin of the turkey when it is done is approximately \(414.6^{\circ} \mathrm{F}\). \((c)\) The total amount of heat transferred to the turkey in the oven is approximately \(1971.08\mathrm{~Btu}\). The thermometer reading will be less than \(185^{\circ} \mathrm{F}\), approximately \(183.5^{\circ} \mathrm{F}\), 5 minutes after the turkey is taken out of the oven.