Chapter 13: Problem 13

Find the steady-state temperature distribution in a rectangular plate covering the area \(0

Short Answer

Expert verified

The steady-state temperature distribution is given by \( T(x, y) = xy \).

Step by step solution

Understand the Problem

The rectangular plate is defined over the area \(0 < x < 10\) and \(0 < y < 20\). Two adjacent sides are maintained at temperatures given by \(T=x\) and \(T=y\), while the other two sides are held at \(0^{\text{o}}\) (zero degrees). The goal is to find the steady-state temperature distribution in the plate.

Identify Boundary Conditions

The boundary conditions can be defined as follows:1. Along the side where \(x=0\): \(T(0,y)=y\)2. Along the side where \(y=0\): \(T(x,0)=x\)3. Along the side where \(x=10\): \(T(10,y)=0\)4. Along the side where \(y=20\): \(T(x,20)=0\)

Set up Laplace's Equation

The temperature distribution \(T(x,y)\) in the steady state must satisfy Laplace’s equation given by \[ \frac{∂^{2} T}{∂x^{2}} + \frac{∂^{2} T}{∂y^{2}} = 0 \]

Assume Solution Form

Assume a solution in the form: \(T(x,y)=X(x)Y(y)\)Substituting into Laplace's equation gives: \[X''(x)Y(y) + X(x)Y''(y) = 0 \]By separating variables, we have: \[\frac{X''(x)}{X(x)} = - \frac{Y''(y)}{Y(y)} = k \]where \(k\) is a separation constant. This provides two ordinary differential equations: \[X''(x) - kX(x) = 0 \]and \[Y''(y) + kY(y) = 0 \]

Solve the Differential Equations

Solving the equations:For \(X(x)\): \[ X(x) = A \, cos( \, \sqrt{k} x \,) + B \, sin( \, \sqrt{k} x \,) \]For \(Y(y)\): \[ Y(y) = C \, cosh( \, \sqrt{k} y \,) + D \, sinh( \, \sqrt{k} y) \]

Apply Boundary Conditions

Using the boundary conditions:At \(x = 0\), \(T(0,y) = y\)\(\Rightarrow C \, cosh(0) + D \, sinh(0) = y\)\(\Rightarrow C = y\)Substituting back:\(Y(y) = y\)At \(y = 0\), \(T(x,0) = x\)\(\Rightarrow A \, cos(0) + B \, sin(0) = x\) \(\Rightarrow A = x\)Substituting back: \(X(x) = x\)

Combine and Simplify the Solution

Combine the solutions of \(X(x)\) and \(Y(y)\) to get:\(T(x, y) = X(x) Y(y) = x \, y\)

Verify the Solution

Check that the solution satisfies all boundary conditions and Laplace's equation.The solution \(T(x, y) = xy\) indeed meets the boundary conditions where \(T(0, y) = y\), \(T(x, 0) = x\), \(T(10, y) = 0\), and \(T(x, 20) = 0\).

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Key Concepts

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

Laplace's equation

Laplace's equation is crucial in solving steady-state problems, such as finding the temperature distribution on a rectangular plate. It is a second-order partial differential equation given by:

For our specific problem, we are working within the domain
which represents a steady-state scenario. Here, temperature does not change over time, implying that the time derivative vanishes. We solve this equation to obtain the temperature distribution in the entire domain.

Boundary Conditions

Boundary conditions are the given values of the function (in this case, temperature) or its derivatives on the boundary of the domain. They are crucial for solving partial differential equations like Laplace's equation. For our problem:

Along the side where x=0:

Along the side where y=0:

Along the side where x=10:

Along the side where y=20:

Functionally, these boundary conditions are used during the solution process to determine specific constants or functions, ensuring that the solution is compatible with the given conditions at the edges of the domain.

Rectangular Plate

For this problem, we deal with a rectangular plate defined over the area

This rectangular domain simplifies the problem by providing straight, parallel edges, making it easier to apply boundary conditions and solve Laplace's equation compared to more complicated shapes.
The temperature distribution throughout the plate depends on the temperatures maintained at the edges, which can vary significantly based on the boundary conditions applied.
Here, two sides are held at known temperatures following the formulas \(T=x\) and \(T=y\), while the other two sides are at zero degrees.
As a result, we can expect a gradual change in temperature from one side of the plate to the other, depending on these boundary conditions.