Question

Find the steady-state temperature distribution in a rectangular plate \(30 \mathrm{cm}\) by \(40 \mathrm{cm}\) given that the temperature is \(0^{\circ}\) along the two long sides and along one short end; the other short end along the \(x\) axis has temperature \(T=\left\\{\begin{array}{cc}100^{\circ}, & 0

Short answer

The steady-state temperature distribution in the rectangular plate is found by solving Laplace's equation with given boundary conditions using separation of variables.

Step-by-step solution

- Define the Problem

Identify the dimensions of the rectangular plate: 30 cm (width, x-direction) by 40 cm (height, y-direction). The temperatures are given as follows: 0°C along the two long sides (y=0 and y=40) and one short end (x=0), and a varying temperature along the other short end (x=30).

- Set Boundary Conditions

Establish the boundary conditions based on the given temperature distribution:\[ T(x, 0) = 0 \text{ for } 0 \text{ cm} \text{ ≤ x ≤ 30} \text{ cm}\]\[ T(x, 40) = 0 \text{ for } 0 \text{ cm} \text{ ≤ x ≤ 30} \text{ cm}\]\[ T(0, y) = 0 \text{ for } 0 \text{ cm} \text{ ≤ y ≤ 40} \text{ cm}\]\[ T(30, y) = 100 \text{ for } 0 \text{ cm} \text{ ≤ y < 10} \text{ cm and } T(30, y) = 0 \text{ for } 10 \text{ cm} \text{ ≤ y ≤ 30} \text{ cm} \]

- Solve the Heat Equation

Use the heat equation for steady-state conditions, which simplifies to Laplace's equation: \[ \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} = 0 \].Plug in the boundary conditions and solve for the temperature distribution T(x, y). This generally involves techniques such as separation of variables and Fourier series.

- Apply Separation of Variables

Express the temperature as a product of two functions, one in terms of x and one in terms of y: \[ T(x, y) = X(x)Y(y) \].Substitute this into Laplace's equation to get: \[ \frac{X''(x)}{X(x)} + \frac{Y''(y)}{Y(y)} = 0 \].Separate into two ordinary differential equations (ODEs): \[ X''(x) = -\lambda X(x) \]\[ Y''(y) = \lambda Y(y) \].

- Solve the Differential Equations

The solutions to these ODEs are:\[ X(x) = A\cos(\sqrt{\lambda}x) + B\sin(\sqrt{\lambda}x)\]\[ Y(y) = C\cosh(\sqrt{\lambda}y) + D\sinh(\sqrt{\lambda}y) \].

- Apply Boundary Conditions to Solutions

Apply the boundary conditions to solve for constants A, B, C, and D, and to find the specific form of the general solution. This involves fitting the boundary temperature values and mode shapes to the problem.

- Construct the General Solution

Combine the specific solutions for each boundary condition to create the full temperature distribution: \[ T(x, y) = \sum_{n=1}^{\infty} A_n \sin\left(\frac{n\pi x}{30}\right)\sinh\left(\frac{n\pi y}{30}\right) \].Determine the constants \(A_n\) by solving the boundary conditions at x=30 cm with the given temperature.

- Final Temperature Distribution

The final solution is the steady-state temperature distribution in the rectangular plate considering all boundary conditions. By fitting boundary conditions, the specific series coefficients can be identified.

Key concepts

Laplace's Equation

Laplace's equation is a second-order partial differential equation that describes the behavior of scalar fields such as temperature or electric potential in the absence of sources. It is written as: \[abla^2 T = \frac{\text{∂}^2 T}{\text{∂} x^2} + \frac{\text{∂}^2 T}{\text{∂} y^2} = 0 \]. This equation is of great importance in steady-state problems, where the system does not change over time. This is because, at steady-state, the rate of change of temperature (or other quantities) is zero. For our problem with the rectangular plate, we use Laplace's equation to find the steady-state temperature distribution. Laplace's equation simplifies the more general heat equation when it assumes steady-state conditions, removing the time-dependent term.

Boundary Conditions

Boundary conditions specify the values that a solution to a differential equation must satisfy on the boundaries of the domain. They are crucial in solving Laplace's equation as they determine the specific solution to a problem. For our exercise, the rectangular plate has the following boundary conditions:

* T(x, 0) = 0 for the bottom edge
* T(x, 40) = 0 for the top edge
* T(0, y) = 0 for the left edge
* T(30, y) = a varying temperature with T(30, y) being 100°C for 0 < y < 10 and 0°C for 10 < y < 30

These boundary conditions help us set up specific functions that represent the temperature distribution on each edge, which guides us in solving Laplace's equation.

Separation of Variables

Separation of variables is a mathematical technique used to solve partial differential equations by breaking them into simpler, one-dimensional problems. We assume the solution to Laplace's equation can be written as a product of two independent functions: one only in terms of x and the other only in terms of y. For example, T(x, y) = X(x)Y(y). Substituting T(x, y) into Laplace's equation separates it into two ordinary differential equations (ODEs):

\[ \frac{X''(x)}{X(x)} + \frac{Y''(y)}{Y(y)} = 0 \] This can be split into:
\[ X''(x) = -\text{λ} X(x) \]
\[ Y''(y) = \text{λ} Y(y) \]

Where λ is a separation constant. This method allows us to handle complex boundary conditions and simplifies the process of finding the temperature distribution.

Fourier Series

A Fourier series is a way to represent a function as the sum of simple sine and cosine waves. By expressing the temperature distribution in terms of Fourier series, we can break down complex boundary conditions into sums of simple, periodic functions. For the rectangular plate, after applying separation of variables, the temperature distribution can be represented as a Fourier series:

\[ T(x, y) = \text{∑}_{n=1}^{\text{∞}} A_n \text{sin}\bigg(\frac{nπx}{30}\bigg) \text{sinh}\bigg(\frac{nπy}{30}\bigg) \]

The constants A_n can be determined by fitting the boundary conditions. This series representation simplifies the calculation and application of various boundary conditions in the solution.

Heat Equation

The heat equation describes how heat (temperature) diffuses through a given region over time. It is given by the partial differential equation:

\[ \frac{\text{∂}T}{\text{∂}t} = \text{α} \bigg( \frac{\text{∂}^2 T}{\text{∂} x^2} + \frac{\text{∂}^2 T}{\text{∂} y^2} \bigg) \]

Where α is the thermal diffusivity constant. For steady-state conditions, the time-dependent term (\frac{\text{∂}T}{\text{∂}t}) on the left-hand side becomes zero, simplifying the heat equation to Laplace's equation:

\[ \frac{\text{∂}^2 T}{\text{∂} x^2} + \frac{\text{∂}^2 T}{\text{∂} y^2} = 0 \]

In our exercise, we solve this simplified version to find the steady-state temperature distribution in the plate.