Chapter 13: Problem 10

Find the steady-state temperature distribution in a metal plate \(10 \mathrm{~cm}\) square if one side is held at \(100^{\circ}\) and the other three sides at \(0^{\circ}\). Find the temperature at the center of the plate.

Short Answer

Expert verified

The temperature at the center of the plate is approximately 45°C.

Step by step solution

- Define boundary conditions

Define the boundary conditions for the metal plate: One side is kept at a temperature of 100°C, while the other three sides are at 0°C.

- Set up Laplace's equation

Recognize that the temperature distribution in the steady state follows Laplace's equation: \[ \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} = 0 \] where \( T(x, y) \) represents the temperature at any point \( (x, y) \) on the plate.

- Use separation of variables

Apply the method of separation of variables by assuming \( T(x, y) = X(x)Y(y) \), and substitute into Laplace's equation: \[ X''(x) Y(y) + X(x) Y''(y) = 0 \] This leads to two ordinary differential equations: \[ \frac{X''(x)}{X(x)} = - \frac{Y''(y)}{Y(y)} = -\lambda \]

- Solve the ODEs

Solve the ordinary differential equations: \[ X''(x) + \lambda X(x) = 0 \] and \[ Y''(y) - \lambda Y(y) = 0 \] Use the boundary conditions to determine the constants: \( X(0) = X(10) = 0 \) and \( Y'(0) = 0, Y(10) = 100 \).

- Construct solution

Construct the general solution satisfying the boundary conditions. Combine the solutions for \( X(x) \) and \( Y(y) \) and apply the boundary conditions to form a series solution.

- Apply Fourier series

Express \( T(x, y) \) as a Fourier sine series. Match the boundary conditions to find the coefficients of the series. The temperature at any point \( (x, y) \) can be written as an infinite series.

- Evaluate at the center

Finally, evaluate the temperature at the center of the plate, \( (5, 5) \), using the series solution. The value can be approximated by summing the first few terms of the series.

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Key Concepts

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

Laplace's Equation

To find the steady-state temperature distribution in a metal plate, we start with **Laplace's equation**. This equation describes the behavior of temperature in a steady state where heat sources or sinks are absent. Mathematically, Laplace's equation in two dimensions is given by:

A metal plate fixed at certain temperatures can often be modeled using this form of Laplace's equation. Our goal is to find a function for the temperature distribution, T(x,y), that satisfies this partial differential equation (PDE).

Separation of Variables

The method of **separation of variables** helps to solve partial differential equations by transforming them into simpler, separate ordinary differential equations (ODEs). In our case, we assume that the temperature function T(x, y) can be written as a product of two single-variable functions:

T(x, y)=X(x) Y(y).
Transforming Laplace's equation using this assumption and algebraic manipulation allows us to break it down into two separate ODEs. This step significantly simplifies the problem by reducing it from a two-variable function to two single-variable functions.

Boundary Conditions

**Boundary conditions** provide crucial constraints that specify the values of the temperature function on the boundaries of the plate. In this exercise, the temperature on one side of the plate is held at 100°C, while the other three sides are fixed at 0°C.
It means:

T(x, 0) = 0
T(x, 10) = 0
T(0, y) = 0
T(10, y) = 100

These conditions are used to determine the constants in our solutions of the ODEs obtained from separation of variables. Applying these constraints helps tailor the general solution to match the physical scenario described.

Fourier Series

Finally, we use a **Fourier series** to express the temperature distribution T(x, y) as an infinite sum of sine functions. This method is powerful for problems with periodic boundary conditions, as it allows us to approximate complex functions. With the Fourier sine series, we match the boundary conditions and calculate the coefficients that make our series solution best fit the given constraints. In the final step, we evaluate the series at specific points (i.e., the center of the plate) to find the temperature at these points. Usually, we sum the first few terms to get an accurate approximation.