Question

Consider a 7.6-cm-long and 3-cm-diameter cylindrical lamb meat chunk \(\left(\rho=1030 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=3.49 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}\right.\), \(\left.k=0.456 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \alpha=1.3 \times 10^{-7} \mathrm{~m}^{2} / \mathrm{s}\right)\). Fifteen such meat chunks initially at \(2^{\circ} \mathrm{C}\) are dropped into boiling water at \(95^{\circ} \mathrm{C}\) with a heat transfer coefficient of \(1200 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). The amount of heat transfer during the first 8 minutes of cooking is (a) \(71 \mathrm{~kJ}\) (b) \(227 \mathrm{~kJ}\) (c) \(238 \mathrm{~kJ}\) \(\begin{array}{ll}\text { (d) } 269 \mathrm{~kJ} & \text { (e) } 307 \mathrm{~kJ}\end{array}\)

Short answer

Answer: (c) 238 kJ

Step-by-step solution

Understand the formula for heat transfer

The formula for the heat transfer (Q) between the meatchunk and boiling water is given by the equation: Q = h * A * ΔT * t where: h = heat transfer coefficient A = surface area ΔT = Temperature difference t = time. We will calculate the parameters based on the given information.

Find the surface area of the meat chunk

The meat is cylindrical in shape. The formula for surface area (A) of a cylinder can be written as: A = 2 * π * r * (r + l) Given: length(l) = 7.6 cm, diameter = 3 cm which means radius(r) = 1.5 cm. Let's find the surface area (A): A = 2 * π * 1.5 * (1.5 + 7.6) = 2 * π * 1.5 * 9.1 = 27.3π cm² Since we want the surface area in meters, we convert the cm² to m²: A = 27.3π × (0.01)² = 0.0273π m²

Find the temperature difference ΔT between the meat chunks and boiling water

Given: initial temperature of meat = 2°C, temperature of the boiling water = 95°C Temperature Difference (ΔT) = Temperature of the boiling water - initial temperature of meat ΔT = 95 - 2 = 93°C

Find the time in seconds

Given: time t = 8 minutes. We need to convert the time to seconds. t = 8 × 60 = 480 seconds

Calculate the heat transfer Q for one meat chunk

Now we have all the parameters to calculate the heat transfer (Q) for one meat chunk: h = 1200 W/m²K A = 0.0273π m² ΔT = 93°C t = 480 seconds Q = h * A * ΔT * t Q = 1200 * 0.0273π *93 * 480 Q = 1200 * 0.0273π * 93 *480 ≈ 15.872 kJ.

Calculate the heat transfer Q for all 15 meat chunks

Since there are 15 meat chunks dropped into the boiling water, we need to multiply the heat transfer for one chunk by 15 to get the total heat transfer: Q(total) = Q * 15 Q(total) = 15.872 * 15 ≈ 238.08 kJ Now we will compare the obtained value with the given options. a) 71 kJ b) 227 kJ c) 238 kJ d) 269 kJ e) 307 kJ The closest option to our calculated value of heat transfer is (c) 238 kJ. So, the heat transfer during the first 8 minutes of cooking is approximately 238 kJ.

Key concepts

Cylindrical Object

In this exercise, we are dealing with a cylindrical object, which is an important shape in various real-world applications and scientific problems. Cylindrical objects have straight sides and circular bases and tops, which makes their physical properties interesting and often more complex to calculate than simple flat surfaces. The cylindrical shape is characterized primarily by its radius and height (or length). In our specific problem, we have a chunk of meat shaped like a cylinder with a radius of 1.5 cm and a length of 7.6 cm. Understanding these dimensions is crucial because they impact calculations like surface area and volume, both of which are key to determining how substances like heat transfer through the object. This basic grasp helps by also figuring out how other phenomena like pressure and force distribute across a cylindrical object.

Surface Area Calculation

The calculation of the surface area of a cylindrical object plays a vital role in determining how much of an object is exposed to the outer environment. For a cylinder, the surface area includes the circular top and bottom as well as the curved surface that connects them. The formula to calculate the surface area (A) of a cylinder is given by:

• \[ A = 2\pi r (r + l) \]
• where \(r\) is the radius and \(l\) is the length of the cylinder

For our cylindrical meat chunk, given the radius as 1.5 cm and length as 7.6 cm, substituting in for the formula results in a surface area calculation of 27.3π cm². However, calculations often require units to be consistent; hence converting cm² to m² (the unit required for our heat transfer formula) gives us 0.0273π m². Calculating accurate surface areas is essential because it affects how much heat or other types of transfer can happen between the object and its surroundings.

Temperature Difference

Temperature difference, denoted as \(\Delta T\), is a crucial element in calculating heat transfer. It determines the potential for energy transfer - specifically, heat - from one body to another. In our scenario, we have a meat chunk initially at 2°C being submerged into boiling water at 95°C. The temperature difference drives the heat transfer process, calculated as:

• \(\Delta T = T_{\text{boiling water}} - T_{\text{initial meat}}\)
• which simplifies to \(\Delta T = 95 - 2 = 93°C\)

The larger the temperature difference, the more intense the rate of heat transfer, assuming all other conditions such as surface area and heat transfer coefficient remain constant. This understanding is crucial when looking into systems where thermal regulation is vital, like cooking, refrigeration, or even in industrial processes.

Time Conversion

Time conversion is a fundamental step often required in physics and engineering calculations because the standard unit for time in these contexts is seconds. Our exercise initially provides the cooking time as 8 minutes. To use time in mathematical formulas, especially those involving rates like heat transfer, we need to convert this time into seconds:

• \(t = 8 \, \text{minutes} \times 60 \, \frac{\text{seconds}}{\text{minute}} = 480 \, \text{seconds}\)

This conversion ensures that calculations remain consistent and accurate. When using formulas that involve time, ensuring the correct unit is essential to obtaining the correct result. This step highlights the importance of attention to detail in scientific calculations, as even minor mistakes in unit conversions can lead to significant errors in outcomes.