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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

01

Set up the boundary conditions

The plate is defined by the region where 0 < x < 10 0 < y < 20 . We have the following boundary conditions: Along the sides where x=0 and x=10, the temperatures are T=y and T=0 respectively. Along the sides where y=0 and y=20, the temperatures are T=x and T=0 respectively.
02

Write the general solution for the Laplace equation

Using separation of variables, the general solution in two dimensions for the steady-state heat equation ∇²T = 0 can be written as T(x,y) = X(x)Y(y).
03

Apply boundary conditions to solve for X(x) and Y(y)

Applying the boundary condition at x=0 (T=y) and x=10 (T=0) tells us: X(0)Y(y)=y X(10)Y(y)=0 This suggests a linear variation for X(x), hence X(x)=A(x)=x.
04

Solve for the solution

Substitute X(x)=x: T(x,y) = xY(y) Using the boundary condition T=0 along y=20 we get: xY(20)=0, thus Y(20)=0 This implies a linear variation for Y(y), hence Y(y)=y, and T(x,y) = xy.

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Key Concepts

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

Laplace Equation
The Laplace Equation is a second-order partial differential equation named after the mathematician Pierre-Simon Laplace. It is written as \( abla^2 T = 0 \), where \( abla^2 \) is the Laplacian operator. This equation is fundamental in the study of steady-state temperature distributions because it describes scenarios where there are no internal heat sources or sinks.

In the context of the exercise, solving the Laplace equation helps us determine the temperature distribution on the plate, assuming heat has evenly diffused across the surface. The two-dimensional version of this equation is \( \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} = 0 \). This simplifies temperature distribution analysis by breaking it down into more manageable calculations using boundary conditions and specific methods like separation of variables.
Boundary Conditions
Boundary conditions are constraints necessary to solve differential equations like the Laplace Equation. They represent the physical conditions or constraints at the edges of the region we are studying. For example, in the exercise, the temperatures along the edges of the plate are specified.

The boundary conditions for the given rectangular plate are:
- Along \( x=0 \): \( T=y \)
- Along \( x=10 \): \( T=0 \)
- Along \( y=0 \): \( T=x \)
- Along \( y=20 \): \( T=0 \)
These conditions are crucial for determining the functions that satisfy the Laplace Equation within the region. By applying these conditions, we are able to constrain possible solutions and find the exact temperature distribution.
Separation Of Variables
Separation of Variables is a powerful method for solving partial differential equations like the Laplace Equation. The idea is to assume that the function we are looking for can be written as a product of functions, each depending on a single variable.

In our exercise, we assume that \( T(x,y) = X(x)Y(y) \), where \( X \) is a function of \( x \) and \( Y \) a function of \( y \). This simplifies the two-dimensional problem into two one-dimensional problems. Applying the boundary conditions progressively yields:
- \( X(x) = x \)
- \( Y(y) = y \)

Thus, the solution to the two-dimensional Laplace Equation is the product \( T(x,y) = xy \).
Steady-State Heat Equation
The steady-state heat equation is another term for the Laplace Equation when used to describe heat distribution without time variation. This means the system has reached a point where temperature does not change over time, only in space.

For the plate, we assume it's in a steady state (no heat flow over time), which leads to \( abla^2 T = 0 \). This simplifies analysis and allows us to focus on spatial temperature variations only. The steady-state solution provides a snapshot of how heat distributes across the plate under the given boundary conditions. For the given exercise, solving the steady-state heat equation with the boundary conditions reveals the final temperature distribution \( T(x,y) = xy \).

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Most popular questions from this chapter

The following two \(R(r)\) equations arise in various separation of variables problems in polar, cylindrical, or spherical coordinates: $$\begin{aligned}&r \frac{d}{d r}\left(r \frac{d R}{d r}\right)=n^{2} R,\\\&\frac{d}{d r}\left(r^{2} \frac{d R}{d r}\right)=l(l+1) R.\end{aligned}$$ There are various ways of solving them: They are a standard kind of equation (often called Euler or Cauchy equations see Chapter \(8,\) Section \(7 \mathrm{d}\) ); you could use power series methods; given the fact that the solutions are just powers of \(r,\) it is easy to find the powers. Choose any method you like, and solve the two equations for future reference. Consider the case \(n=0\) separately. Is this necessary for \(l=0 ?\)

Solve the semi-infinite plate problem if the bottom edge of width \(\pi\) is held at \(T=\cos x\) and the other sides are at \(0^{\circ}\).

Show that there is only one function \(u\) which takes given values on the (closed) boundary of a region and satisfies Laplace's equation \(\nabla^{2} u=0\) in the interior of the region. Hints: Suppose \(u_{1}\) and \(u_{2}\) are both solutions with the same boundary conditions so that \(U=u_{1}-u_{2}=0\) on the boundary. In Green's first identity (Chapter 6, Problem 10.16), let \(\phi=\Psi=U\) to show that \(\nabla U \equiv 0 .\) Thus show \(\bar{U} \equiv 0\) everywhere inside the region.

A sphere initially at \(0^{\circ}\) has its surface kept at \(100^{\circ}\) from \(t=0\) on (for example, a frozen potato in boiling water!). Find the time- dependent temperature distribution. Hint: Subtract \(100^{\circ}\) from all temperatures and solve the problem; then add the \(100^{\circ}\) to the answer. Can you justify this procedure? Show that the Legendre function required for this problem is \(P_{0}\) and the \(r\) solution is \((1 / \sqrt{r}) J_{1 / 2}\) or \(j_{0}\) [see (17.4) in Chapter 12]. since spherical Bessel functions can be expressed in terms of elementary functions, the series in this problem can be thought of as either a Bessel series or a Fourier series. Show that the results are identical.

The Klein-Gordon equation is \(\nabla^{2} u=\left(1 / v^{2}\right) \partial^{2} u / \partial t^{2}+\lambda^{2} u .\) This equation is of interest in quantum mechanics, but it also has a simpler application. It describes, for example, the vibration of a stretched string which is embedded in an elastic medium. Separate the one-dimensional Klein-Gordon equation and find the characteristic frequencies of such a string.
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