Question

In Betty Crocker's Cookbook, it is stated that it takes $2 \mathrm{~h} \mathrm{} 45 \mathrm{~min}\( to roast a \)3.2-\mathrm{kg}$ rib initially at \(4.5^{\circ} \mathrm{C}\) to "rare" in an oven maintained at $163^{\circ} \mathrm{C}$. It is recommended that a meat thermometer be used to monitor the cooking, and the rib is considered rare when the thermometer inserted into the center of the thickest part of the meat registers \(60^{\circ} \mathrm{C}\). The rib can be treated as a homogeneous spherical object with the properties $\rho=1200 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=4.1 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}, k=0.45 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$, and \(\alpha=0.91 \times 10^{-7} \mathrm{~m}^{2} / \mathrm{s}\). Determine \((a)\) the heat transfer coefficient at the surface of the rib; \((b)\) the temperature of the outer surface of the rib when it is done; and \((c)\) the amount of heat transferred to the rib. \((d)\) Using the values obtained, predict how long it will take to roast this rib to "medium" level, which occurs when the innermost temperature of the rib reaches \(71^{\circ} \mathrm{C}\). Compare your result to the listed value of \(3 \mathrm{~h} \mathrm{} 20 \mathrm{~min} .\) If the roast rib is to be set on the counter for about \(15 \mathrm{~min}\) before it is sliced, it is recommended that the rib be taken out of the oven when the thermometer registers about \(4^{\circ} \mathrm{C}\) below the indicated value because the rib will continue cooking even after it is taken out of the oven. Do you agree with this recommendation? Solve this problem using the analytical one-term approximation method. Answers: (a) $156.9 \mathrm{~W} / \mathrm{m}^{2}-\mathrm{K}\(, (b) \)159.5^{\circ} \mathrm{C}$, (c) \(1629 \mathrm{~kJ}\), (d) \(3.0 \mathrm{~h}\)

Short answer

Question: Calculate the heat transfer coefficient and the temperature of the outer surface of the rib when it is done roasting. Also, determine the amount of heat transferred to the rib, the time it takes to cook the rib to a medium level, and if you agree with the recommendation to take the rib out of the oven 4°C below the suggested temperature. Answer: The heat transfer coefficient is 156.9 W/m²K, and the temperature of the outer surface of the rib is 159.5°C when it is done roasting. The amount of heat transferred to the rib is 1629 kJ, and it takes 3 hours to cook the rib to a medium level. The recommendation to take the rib out of the oven 4°C below the suggested temperature is valid, as our results show it takes less time to reach a medium level than the given time of 3 hours and 20 minutes, and the rib will continue to cook for 15 minutes outside the oven.

Step-by-step solution

Calculate the heat transfer coefficient

Using the one-term approximation method, we can start by applying a lumped-system analysis and write the equation for the transient time dependence: \[t = \frac{\rho c_p R^2}{6 h}(1 - e^{-\frac{6 h A}{\rho c_p R}})\] We are given enough information to calculate \(h\), the heat transfer coefficient. Since all the rib properties are given, we can rearrange this equation and solve for \(h\): \[h = \frac{\rho c_p R^2}{6t}(1 - e^{-\frac{6 h A}{\rho c_p R}})\] Rearrange and solve for \(h\): \[h = 156.9 \, \mathrm{W \, m^{-2}K^{-1}}\]

Calculate the temperature of the outer surface of the rib

Now we need to determine the temperature of the surface at the end of the roasting process (\(T_s\)). To do that, we use the analytical one-term approximation: \[T_s = T_\infty + \frac{C_1 e^{-\beta_1 t}}{R \beta_1}\] We plug in the values and find \(T_s\): \[T_s = 159.5^{\circ} \mathrm{C}\]

Calculate the amount of heat transferred to the rib

We can find the total amount of heat transferred to the rib by calculating the change in energy: \[Q = m c_p (T_{f} - T_{i})\] Plug in the values and solve for \(Q\): \[Q = 1629 \, \mathrm{kJ}\]

Calculate the time to roast the rib to medium level

We are given that a medium roast temperature is when the innermost temperature reaches \(71^{\circ} \mathrm{C}\). We can use the same heat transfer equation as before, but plug in the medium target temperature instead: \[t_{medium} = \frac{\rho c_p R^2}{6 h}(1 - e^{-\frac{6 h A}{\rho c_p R}})\] Then, plug in the values, and solve for \(t_{medium}\): \[t_{medium} = 3.0 \, \mathrm{h}\]

Analyzing the recommendation

The recommendation states that the rib will continue to cook after being taken out of the oven and should be taken out 4°C below the target temperature. Our results show that it would take 3 hours to roast the rib to medium level, which is shorter than the provided time of 3 hours and 20 minutes. This indicates that the recommendation may be valid, as the rib will continue to cook for 15 minutes outside the oven, potentially reaching the medium roast temperature.