Problem 46 - Advanced Engineering Mathematics

Question

A small metal bar is removed from an oven whose temperature is a constant \(300^{\circ} \mathrm{F}\) into a room whose temperature is a constant \(70^{\circ} \mathrm{F}\). Simultaneously, an identical metal bar is removed from the room and placed into the oven. Assume that time \(t\) is measured in minutes. Discuss: Why is there a future value of time, call it \(t^{*}>0\), at which the temperature of each bar is the same?

Step-by-Step Solution

Answer

The symmetrical nature of heat exchange under Newton's Law of Cooling ensures a moment exists when both bars have the same temperature.

Step 1: Understanding the Problem

Two identical metal bars are swapped between environments with different temperatures. One goes from the oven (300°F) to the room (70°F), and the other from the room to the oven. We need to determine if there's a moment when both bars have the same temperature.

Step 2: Recognizing Newton's Law of Cooling

Newton's Law of Cooling suggests that the rate of change of temperature of an object is proportional to the difference between its temperature and the surrounding environment's temperature. The formula is usually expressed as \( \frac{dT}{dt} = -k(T - T_{env}) \), where \( T \) is the object's temperature, \( T_{env} \) is the environment's temperature, and \( k \) is a positive constant.

Step 3: Setting the Initial Conditions

For the bar from the oven: initial temperature \( T_1(0) = 300^{\circ} \mathrm{F} \) and environment temperature \( T_{env} = 70^{\circ} \mathrm{F} \). For the bar from the room: initial temperature \( T_2(0) = 70^{\circ} \mathrm{F} \) and environment temperature \( T_{env} = 300^{\circ} \mathrm{F} \).

Step 4: Analyzing temperature change over time

Apply the cooling law to both bars. After some time, the temperature of the bar from the oven decreases towards the room temperature, and the temperature of the bar in the oven increases towards oven temperature. This eventually leads to a point where both temperatures are equal.

Step 5: Applying Symmetry and Physics Reasoning

Consider the symmetry: each bar's temperature moves towards the other's starting environment—room for one, oven for the other. Since the process is continuous and both objects are identical, it's inevitable that temperatures equalize at some point after being swapped.

Step 6: Conclusion

The continuous and opposing nature of the temperature changes, driven by the identical rate constants for both bars and their reverse placements, ensures there is a time \( t^*>0 \) when their temperatures intersect.

Key Concepts

Heat Transfer

Heat transfer is the movement of thermal energy from one object or substance to another. This process occurs due to the temperature difference, much like in our scenario where a metal bar is moved from a hot oven to a cooler room, and vice versa. When two objects are placed in different thermal environments, heat will naturally flow from the hotter object to the cooler one until a state of balance, or thermal equilibrium, is reached.

• Heat naturally flows from hot to cold.
• The greater the temperature difference, the faster the heat transfer.
• Heat transfer will continue until thermal equilibrium is achieved.

In the case of the swapped metal bars, the heat transfer results in the oven bar cooling down and the room bar heating up. Each bar undergoes this process simultaneously, aligning with Newton's Law of Cooling.

Differential Equations

Differential equations are mathematical equations that describe how a quantity changes with the rate of change. In our exercise, Newton's Law of Cooling involves a differential equation. This law states that the rate at which an object cools (or heats up) is proportional to the temperature difference between the object and its environment.

• The equation can be written as: \( \frac{dT}{dt} = -k(T - T_{env}) \), where:
• \( T \) is the temperature of the object.
• \( T_{env} \) is the environmental temperature.
• \( k \) is a positive constant.

Solving this differential equation allows us to predict how the temperature of the metal bars will change over time. By integrating the equation, we can express the temperature of each bar as a function of time. This forms the basis for understanding how and when the temperatures of the two bars will intersect.

Temperature Equilibrium

Temperature equilibrium is a concept where differing temperatures of objects balance out, reaching a common, stable temperature. In our problem, each bar starts at a different temperature: one hot and one cold. Over time, due to heat transfer and Newton's Law of Cooling, their temperatures adjust towards each other.

• Equilibrium occurs when no net heat transfer happens because temperatures are equal.
• In our scenario, equilibrium is reached when both bars have the same temperature simultaneously.

The interplay of cooling and heating processes for each bar ensures that, eventually, a specific time \( t^{*} \) will exist where the temperatures of both bars are equal. This happens when the temperatures have moved sufficiently towards each other's starting environments, and this time can theoretically be determined mathematically.