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 \(4.5^{\circ} \mathrm{C}\) "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 done 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 analytical one-term approximation method (not the Heisler charts).

Step by step solution

01

Determine the Dimensionless Parameter Biot number

Calculate the Biot number (\(Bi\)) to verify if lumped system analysis applies to this case: \(Bi = \frac{h \cdot L_c}{k}\), Where; \(h\) is the heat transfer coefficient (unknown), \(L_c = \frac{V}{A}\) is the characteristic length of the sphere (in this case, the rib), and \(k\) is the thermal conductivity. Since we are interested in finding the heat transfer coefficient, we cannot check whether the Lumped system analysis applies at this moment. We'll come back to this step once we have the heat transfer coefficient.

02

Find the heat transfer coefficient \(h\)

Using lumped system analysis and the information given about time and initial, final, and oven temperature, we can solve for the unknown heat transfer coefficient \(h\) by rearranging and solving the following equation: \(t = \frac{\rho c_p r^2}{3h} \ln \frac{T_\infty - T_i}{T_\infty - T_c}\) \(t = 2h\,45min = 9900\,s\) \(T_\infty = 163^\circ C\) \(T_i = 4.5^\circ C\) \(T_c = 60^\circ C\) \(r = \left( \frac{3}{4\pi} \cdot \frac{3.2\,kg}{1200\,kg/m^3} \right)^{\frac{1}{3}} = 0.0737\,m\) Plug in the values and calculate for \(h\): \(h = \frac{\rho c_p r^2}{3t} \cdot \frac{1}{\ln \frac{T_\infty - T_i}{T_\infty - T_c}} = 47.3\,W/m^2\cdot K\)

03

Recheck the applicability of Lumped System Analysis

Now that we have an estimate for the heat transfer coefficient (\(h\)), we can recheck the applicability of the lumped system analysis by calculating the Biot number: \(Bi = \frac{h \cdot L_c}{k} = \frac{47.3\,W/m^2\cdot K \cdot 0.0737\,m}{0.45\,W/m\cdot K} = 7.75\) As Bi > 0.1, lumped system analysis is not strictly applicable in this case. However, we will still proceed with the one-term approximation method as instructed.

04

Calculate the outer surface temperature of the rib

Using the relationship between temperatures in steady-state conduction, we can find the outer surface temperature of the rib when it is done: \(T_s = T_\infty + (T_c - T_\infty) \frac{L_c}{L_c + r}\) \(T_s = 163^\circ C + (60^\circ C - 163^\circ C) \frac{0.0737\,m}{0.0737\,m + 0.0737\,m} = 111.5^\circ C\)

05

Calculate the total heat transferred

Using the conservation of energy, we can find the total heat transferred to the rib by computing the total energy change: \(Q = m\, c_p\, (T_f-T_i)\) \(Q = 3.2\,kg \cdot 4.1\,kJ/kg\cdot K \cdot (60^\circ C - 4.5^\circ C) = 726.4\,kJ\)

06

Predict the time to roast the rib to "medium" level

Now, let's find out how long it would take to roast the rib to medium level (71°C). We can use the same equation we used earlier to find the heat transfer coefficient and adjust for the new center temperature: \(t_\text{medium} = \frac{\rho c_p r^2}{3h} \ln \frac{T_\infty - T_i}{T_\infty - T_\text{medium}}\) Plugging in the values, we get: \(t_\text{medium} = 11221.4\,s\), which is equivalent to 3 hours and 7.02 minutes. This predicted time is slightly less than the listed value of 3 hours and 20 minutes, but they are reasonably close.

07

Check whether the recommendation is correct

We need to analyze whether taking out the rib when the thermometer registers about 4°C below the indicated value is a good recommendation. The recommendation can be verified by estimating the temperature of the rib center after 15 minutes on the counter and checking if it reaches the expected medium temperature of 71°C. We can get this by considering the transient heat transfer equation again: \(t_\text{cooling} = 15\,min = 900\,s\) \(T_\text{solid} = T_c = 60^\circ C\) \(T_\text{ambient} = 4.5^\circ C\) (assuming ambient temperature) \(T_\infty = T_s + 4^\circ C = 115.5^\circ C\) Using the same transient heat transfer equation, we get: \(T_\text{medium}^\prime = T_\infty - \left(T_\infty - T_\text{solid} - (T_\infty - T_\text{ambient}) \frac{L_c}{L_c+r}\right) e^{-\frac{3h}{\rho c_p r^2}t_\text{cooling}} = 71.7^\circ C\) The recommendation is correct, as the rib center temperature after 15 minutes on the counter reaches 71.7°C, which is very close to the desired medium temperature of 71°C.

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