/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 148 In Betty Crocker's Cookbook, it ... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 148

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?

Short Answer

Expert verified
Answer: The thermometer reading 5 minutes after the turkey is taken out of the oven is approximately 183.5掳F.

Step by step solution

01

Calculate the dimensionless radius and time

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 \(\tau = \frac{\alpha t}{r^2} = \frac{0.0035 \times 5}{(1/3)^2} = 15.75\).
02

Find average heat transfer coefficient at the surface of the turkey

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, \(h \approx 0.918\mathrm{~Btu}/\mathrm{h} \cdot \mathrm{ft}^2 \cdot{ }^{\circ} \mathrm{F}\).
03

Calculate the total amount of heat transferred to the turkey

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 \(1971.08\mathrm{~Btu}\).
04

Calculate the thermometer reading 5 min after taking out the turkey

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Thermal Conductivity
Thermal conductivity is a fundamental concept in heat transfer. It refers to a material's ability to conduct heat. In cooking, understanding thermal conductivity allows us to anticipate how quickly food heats up in the oven.
The higher the thermal conductivity, the faster the heat flows through the material. For instance, a turkey roasting in an oven relies on the thermal conductivity of its meat to evenly distribute heat from the surface to its center.
The turkey in our exercise has been given a thermal conductivity value of \(k = 0.26 \; \text{Btu} / \text{hr} \cdot \text{ft} \cdot{} ^\circ \text{F}\), which means:
  • Heat flows progressively from the turkey's surface towards its core.
  • This property helps achieve a perfectly cooked interior while ensuring the outside doesn鈥檛 burn too quickly.
Brushing up on recipes or preparation methods can influence the effectiveness of thermal conductivity. Tactics like stuffing or letting a turkey rest at room temperature before cooking can alter how quickly heat traverses through the food.
Heat Transfer Coefficient
The heat transfer coefficient is a measure of convective heat transfer between a surface and the fluid around it鈥攊n our case, the turkey and the air in the oven. It quantifies how efficiently heat is transferred from the air onto the surface of the turkey.
A significant part of this exercise was determining the average heat transfer coefficient \(h\), calculated to be approximately \(0.918 \; \text{Btu}/\text{hr} \cdot \text{ft}^2 \cdot{} ^\circ \text{F}\). This value reflects:
  • How quickly the turkey absorbs the oven's warmth at its surface.
  • The thickness of the film of hot air enveloping the turkey, as a thinner film results in a higher coefficient \(h\).
To calculate \(h\), you need to know the thermal conductivity \(k\) of the turkey and its radius \(R\). The formula used is: \[ h = \frac{4}{3} \frac{k}{R} \]This coefficient is crucial because even a recipe with the perfect time and temperature can falter if heat transfer is inefficient. Usage of proper oven settings and preheating techniques are vital to achieve a balanced \(h\).
Cooking Temperature
Cooking temperature plays a vital role in the overall heat transfer during cooking efforts. It is the driving force that pushes heat from the oven into the food, determining how fast and uniformly a turkey cooks. In this example, the oven heat was set at \(325^degree \text{F}\), creating the necessary environment for roasting the \(14-\text{lb}\) turkey uniformly for \(5\) hours.
Several key aspects to consider about cooking temperature are:
  • Higher temperatures cook faster: Increasing the temperature might speed up the process but can sometimes result in an unevenly cooked dish, potentially burning the exterior before the interior is done.

  • Optimal temperature balances speed and quality: For turkeys, temperatures around \(325^degree \text{F}\) are commonly recommended to ensure both a well-cooked inside and a crispy exterior.

  • Thermal equilibrium: The bird warms to its desired internal temperature of \(185^degree \text{F}\) when the heat applied matches the internal heat losses until everything is balanced out.
While the recommended oven temperature is always given, personal modifications can be made based on specific kitchen appliances. It should be noted that roasting at too low temperatures might lead to undercooked outcomes.

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Most popular questions from this chapter

Consider heat transfer between two identical hot solid bodies and their environments. The first solid is dropped in a large container filled with water, while the second one is allowed to cool naturally in the air. For which solid is the lumped system analysis more likely to be applicable? Why?

For heat transfer purposes, an egg can be considered to be a \(5.5-\mathrm{cm}\)-diameter sphere having the properties of water. An egg that is initially at \(8^{\circ} \mathrm{C}\) is dropped into the boiling water at \(100^{\circ} \mathrm{C}\). The heat transfer coefficient at the surface of the egg is estimated to be \(800 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). If the egg is considered cooked when its center temperature reaches \(60^{\circ} \mathrm{C}\), determine how long the egg should be kept in the boiling water. Solve this problem using analytical one-term approximation method (not the Heisler charts).

A long roll of 2-m-wide and \(0.5\)-cm-thick 1-Mn manganese steel plate coming off a furnace at \(820^{\circ} \mathrm{C}\) is to be quenched in an oil bath \(\left(c_{p}=2.0 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}\right)\) at \(45^{\circ} \mathrm{C}\). The metal sheet is moving at a steady velocity of \(15 \mathrm{~m} / \mathrm{min}\), and the oil bath is \(9 \mathrm{~m}\) long. Taking the convection heat transfer coefficient on both sides of the plate to be \(860 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), determine the temperature of the sheet metal when it leaves the oil bath. Also, determine the required rate of heat removal from the oil to keep its temperature constant at \(45^{\circ} \mathrm{C}\).

Consider two identical 4-kg pieces of roast beef. The first piece is baked as a whole, while the second is baked after being cut into two equal pieces in the same oven. Will there be any difference between the cooking times of the whole and cut roasts? Why?

The walls of a furnace are made of \(1.2\)-ft-thick concrete \(\left(k=0.64 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft} \cdot{ }^{\circ} \mathrm{F}\right.\) and \(\left.\alpha=0.023 \mathrm{ft}^{2} / \mathrm{h}\right)\). Initially, the furnace and the surrounding air are in thermal equilibrium at \(70^{\circ} \mathrm{F}\). The furnace is then fired, and the inner surfaces of the furnace are subjected to hot gases at \(1800^{\circ} \mathrm{F}\) with a very large heat transfer coefficient. Determine how long it will take for the temperature of the outer surface of the furnace walls to rise to \(70.1^{\circ} \mathrm{F}\). Answer: \(116 \mathrm{~min}\)
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