Problem 6 - Advanced Engineering Mathematics

Question

A rod of length \(L\) coincides with the interval \([0, L]\) on the \(x\) -axis. Set up the boundary-value problem for the temperature \(u(x, t)\) The ends are insulated, and there is heat transfer from the lateral surface of the rod into the surrounding medium held at temperature \(50^{\circ} .\) The initial temperature is \(100^{\circ}\) throughout.

Step-by-Step Solution

Answer

The boundary-value problem consists of the heat equation with Newton's cooling terms, insulated boundaries, and initial condition.

Step 1: Identify the Physical Situation

We have a rod of length \(L\) located on the \(x\)-axis from \(x = 0\) to \(x = L\). We need to set up the boundary-value problem to describe the temperature \(u(x, t)\) at any location \(x\) and time \(t\). The rod ends are insulated, meaning no heat flows through \(x=0\) and \(x=L\). Additionally, heat can transfer from the lateral surface to the surrounding medium, which is at \(50^{\circ}C\). Initially, the temperature along the rod is \(100^{\circ}C\).

Step 2: Write the Heat Equation

The heat conduction in the rod can be modeled by the one-dimensional heat equation:\[\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2},\]where \(u(x, t)\) is the temperature at position \(x\) and time \(t\), and \(\alpha\) is the thermal diffusivity of the rod material.

Step 3: Apply Boundary Conditions

Since the ends \(x=0\) and \(x=L\) are insulated, we apply the Neumann boundary conditions:\[\frac{\partial u}{\partial x}(0, t) = 0, \quad \frac{\partial u}{\partial x}(L, t) = 0.\]

Step 4: Incorporate Lateral Heat Transfer

To account for heat transfer from the rod to the surrounding medium, we assume it follows Newton's Law of Cooling. This creates additional terms in the form of source sink in the equation:\[- a(u(x, t) - 50),\]where \(a\) is the heat transfer coefficient.

Step 5: Combine Information into a Full Problem Statement

Combining the heat equation, boundary conditions, and lateral heat transfer term, the boundary-value problem becomes:\[\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2} - a(u(x, t) - 50),\]subject to\[\frac{\partial u}{\partial x}(0, t) = 0, \quad \frac{\partial u}{\partial x}(L, t) = 0,\]and with the initial condition:\[u(x, 0) = 100.\]

Key Concepts

Heat Equation

The heat equation is a partial differential equation that describes how heat diffuses through a given region over time. It expresses the change of temperature with both respect to time and space, essentially modeling the process of heat conduction. For a rod along the x-axis, this equation is given by:\[\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}\]This equation tells us how the temperature of the rod, represented by \(u(x, t)\), evolves over time \(t\) and location \(x\). The parameter \(\alpha\) is called the thermal diffusivity and determines how quickly heat spreads through the material.
This simple and powerful equation helps predict temperature changes in a system, which is essential for designing materials and systems in engineering and science.
To fully solve the heat equation for a specific physical scenario, like our rod, we need to specify initial and boundary conditions that reflect how the rod is held and how the temperature behaves at its boundaries and initially.

Neumann Boundary Condition

Boundary conditions are an integral part of solving differential equations like the heat equation, especially when defining a physical system. Neumann boundary conditions specify the derivative of a function at a boundary.
For our rod, the Neumann boundary condition is critical because its ends are insulated. This means no heat enters or leaves through these ends (0 and \(L\)). Mathematically, this is expressed as:\[\frac{\partial u}{\partial x}(0, t) = 0, \quad \frac{\partial u}{\partial x}(L, t) = 0\]These equations imply that the rate of temperature change in the direction of the rod's length is zero at both ends. Therefore, heat only moves through the lateral surfaces of the rod, affecting the temperature changes internally over time. Understanding and applying Neumann boundary conditions ensure that the modeled temperature accurately reflects reality, as no external heat affects the rod's insulated ends.

Newton's Law of Cooling

Newton's Law of Cooling describes how the temperature of an object changes as it exchanges heat with its surroundings. In the case of our rod, the lateral surface exchanges heat with the surrounding medium, with a constant temperature of \(50^{\circ}C\).
This law is modeled by the term:\[- a(u(x, t) - 50),\]where \(a\) is the heat transfer coefficient. This coefficient measures how effectively the rod exchanges heat with its environment. The term \(u(x, t) - 50\) represents the difference in temperature between the rod and the environment. If the rod's temperature \(u(x, t)\) is greater than \(50^{\circ}C\), it loses heat; otherwise, it gains heat.
This element is important because it adds to the complexity of solving the heat equation, accounting for the rod's interaction with its environment. By incorporating Newton's Law of Cooling, our model can more accurately reflect the dynamics of heat transfer in real-life situations, ensuring effective predictions and solutions.