Problem 42 - Thomas Calculus

Question

A beam of unknown temperature An aluminum beam was brought from the outside cold into a machine shop where the temperature was held at \(65^{\circ} \mathrm{F}\) . After 10 min, the beam warmed to \(35^{\circ} \mathrm{F}\) and after another 10 \(\mathrm{min}\) it was \(50^{\circ} \mathrm{F}\) . Use Newton's Law of Cooling to estimate the beam's initial temperature.

Step-by-Step Solution

Answer

The beam's initial temperature was approximately \(5^{\circ} \mathrm{F}\).

Step 1: Understand Newton's Law of Cooling

Newton's Law of Cooling states that the rate of change of temperature of an object is proportional to the difference between its temperature and the ambient temperature. Mathematically, it is given by \( \frac{dT}{dt} = -k(T - T_{ambient}) \), where \(T\) is the temperature of the object, \(T_{ambient}\) is the ambient temperature, and \(k\) is a positive constant.

Step 2: Express Temperature Equation

The solution to the differential equation from Newton's Law of Cooling is \( T(t) = T_{ambient} + (T_{initial} - T_{ambient})e^{-kt} \), where \( T(t) \) is the temperature at time \( t \), \( T_{initial} \) is the initial temperature, and \( e \) is the base of the natural logarithm.

Step 3: Set Up Equations for Given Conditions

We know that \( T_{ambient} = 65 \), \( T(10) = 35 \), and \( T(20) = 50 \). Substituting these into our equation gives us two equations: 1. \( 35 = 65 + (T_{initial} - 65)e^{-10k} \) 2. \( 50 = 65 + (T_{initial} - 65)e^{-20k} \).

Step 4: Solve for k Using Two Equations

From Equation 1, we have \( 35 - 65 = (T_{initial} - 65)e^{-10k} \) leading to \( -30 = (T_{initial} - 65)e^{-10k} \). From Equation 2, \( 50 - 65 = (T_{initial} - 65)e^{-20k} \) simplifies to \( -15 = (T_{initial} - 65)e^{-20k} \). Now, divide the second equation by the first: \( \frac{-15}{-30} = \frac{e^{-20k}}{e^{-10k}} \), which simplifies to \( \frac{1}{2} = e^{-10k} \). Solve for \( k \): \( 10k = \,ln(2) \) => \( k = \frac{\ln(2)}{10} \).

Step 5: Determine Initial Temperature

Use the value of \( k \) found in Step 4 and substitute it back into one of the original equations. Using \( 35 = 65 + (T_{initial} - 65)e^{-10k} \), substitute \( e^{-10k} = \frac{1}{2} \) to get:\( 35 = 65 + (T_{initial} - 65) \cdot \frac{1}{2} \). Solve for \(T_{initial}\): \( 35 - 65 = \frac{1}{2}(T_{initial} - 65) \) leading to \( -30 = \frac{1}{2}(T_{initial} - 65) \) \( T_{initial} - 65 = -60 \), \( T_{initial} = 5 \).

Step 6: Finalize Solution

The initial temperature of the aluminum beam was \( 5^{\circ} \mathrm{F} \). We solved this using the exponential decay relationship derived from Newton's Law of Cooling.

Key Concepts

Differential Equation

A differential equation is a mathematical equation that involves the derivatives of a function. It describes how a quantity changes over time or space. For Newton's Law of Cooling, the key is to understand that the rate of temperature change depends on the difference between the current temperature of the object and the ambient temperature. The equation is written as \[ \frac{dT}{dt} = -k(T - T_{ambient}) \]where:

• \( T \) is the temperature of the object at time \( t \).
• \( T_{ambient} \) is the surrounding temperature, which remains constant.
• \( k \) is a positive constant that represents the rate at which the object is cooling (or warming).

The negative sign in the differential equation indicates that the temperature of the object will decrease over time if it is initially hotter than the ambient temperature. Conversely, the temperature will increase if the object is initially cooler. The solution to this differential equation helps us to predict how the temperature of an object evolves over time, which is essential in solving real-world problems like the one we have with the aluminum beam.

Temperature Modeling

Temperature modeling involves creating mathematical models to predict how the temperature of an object changes over time. In the case of Newton's Law of Cooling, the model derived from the differential equation provides a precise way to calculate temperature at any given point in time. The solution to the differential equation is:\[ T(t) = T_{ambient} + (T_{initial} - T_{ambient})e^{-kt} \]This formula allows us to model how the temperature changes by inputting different values:

• \( T(t) \) represents the temperature of the object at time \( t \).
• \( T_{initial} \) is the starting temperature of the object.
• \( e^{-kt} \) is the exponential decay, indicating how quickly the temperature approaches the ambient temperature.

The exponential decay factor \( e^{-kt} \) shows the characteristic behavior of cooling or warming processes, representing how quickly the temperature difference shrinks over time. This model helps in accurately predicting and understanding temperature changes in various environmental conditions.

Heat Transfer

Heat transfer is the movement of thermal energy from one object or material to another. It can occur through different mechanisms such as conduction, convection, or radiation. Newton's Law of Cooling focuses on heat transfer by conduction and convection, as it primarily deals with how an object's temperature changes when placed in a different thermal environment.

Here, the temperature difference between the object and its surrounding environment is what drives the heat transfer process. As this temperature difference decreases, the rate of heat transfer also diminishes. This concept is what underlies the exponential decay in the temperature modeling equation. The essence of Newton's Law of Cooling is that the greater the temperature difference, the faster the rate of heat transfer up to a certain point.

Understanding heat transfer is crucial in various engineering applications, such as designing heating and cooling systems, predicting energy consumption, and improving machine efficiency. By grasping how heat transfer affects temperature changes, we can make informed decisions in scientific and industrial processes.