Question

Find the steady-state temperature distribution for the semi-infinite plate problem if the temperature of the bottom edge is \(T=f(x)=x\) (in degrees; that is, the temperature at \(x \mathrm{cm}\) is \(x\) degrees), the temperature of the other sides is \(0^{\circ},\) and the width of the plate is \(10 \mathrm{cm}\).

Short answer

The steady-state temperature distribution is given by \[ T(x, y) = \sum_{n=1,3,5,\ldots} \frac{80}{(n \pi)^2} \sin \left( \frac{n \pi x}{10} \right) e^{-\frac{n \pi y}{10}} \].

Step-by-step solution

- Understand the Boundary Conditions

The plate is semi-infinite with a width of 10 cm. The temperature distribution is specified as follows: the bottom edge has a temperature distribution given by \( T = f(x) = x \) for \( 0 \text{ cm} \leq x \leq 10 \text{ cm} \). The temperature along the other three edges (sides and top) is \( 0^\text{C} \).

- Set Up the Governing Equation

The steady-state temperature distribution for a two-dimensional heat conduction problem is governed by Laplace's equation, \( abla^2 T = 0 \). In Cartesian coordinates, this becomes: \[ \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} = 0 \]

- Use Separation of Variables

Assume a solution of the form \( T(x, y) = X(x)Y(y) \). Substituting this into Laplace's equation gives: \[ Y(y) \frac{d^2 X}{dx^2} + X(x) \frac{d^2 Y}{dy^2} = 0 \] Divide both sides by \( X(x)Y(y) \): \[ \frac{1}{X(x)} \frac{d^2 X}{dx^2} + \frac{1}{Y(y)} \frac{d^2 Y}{dy^2} = 0 \] Each term must be equal to a constant, say \( -k^2 \), leading to two ordinary differential equations.

- Solve the Ordinary Differential Equations

The resulting equations are: \[ \frac{d^2 X}{dx^2} + k^2 X = 0 \] and \[ \frac{d^2 Y}{dy^2} - k^2 Y = 0 \]. Solutions to these equations are: \[ X(x) = A \sin(kx) + B \cos(kx) \] \[ Y(y) = C e^{ky} + D e^{-ky} \].

- Apply Boundary Conditions

Apply the boundary conditions to determine constants: 1) \( T(x, 0) = x \implies B Y(0) = x \) for \( 0 \leq x \leq 10 \), 2) \( T(0, y) = T(10, y) = 0 \implies X(0) Y(y) = 0 \) and \( X(10) Y(y) = 0 \), and 3) \( T(x, \infty) = 0 \implies C = 0 \).

- Determine Constants from Boundary Values

Since \( T(0, y) = 0 \), for non-trivial solutions, \( B = 0 \). From \( X(10) = A \sin(k \cdot 10) = 0 \), it follows that \( k_n = \frac{n \pi}{10} \) for \( n = 1, 2, 3, \ldots \). The general solution combining all modes is: \[ T(x, y) = \sum_{n=1}^{\infty} \sin \left( \frac{n \pi x}{10} \right) e^{-\frac{n \pi y}{10}} \]

- Match Boundary Condition at the Bottom Edge

Using Fourier series expansion for \( f(x) = x \): \[ T(x, 0) = x = \sum_{n=1}^{\infty} a_n \sin \left( \frac{n \pi x}{10} \right) \]. Coefficients \( a_n \) are found by: \[ a_n = \frac{2}{10} \int_0^{10} x \sin \left( \frac{n \pi x}{10} \right) dx = \frac{40}{(n \pi)^2} (1 - (-1)^n) \]. Only odd terms contribute.

- Final Solution

Combining all terms, the steady-state temperature distribution is: \[ T(x, y) = \sum_{n=1,3,5,\ldots} \frac{80}{(n \pi)^2} \sin \left( \frac{n \pi x}{10} \right) e^{-\frac{n \pi y}{10}} \]

Key concepts

Laplace's equation

Laplace's equation is a fundamental equation in mathematics and physics used to describe the behavior of scalar fields like temperature, potential, and pressure at steady-state. In two dimensions, it is expressed as: \[ abla^2 T = \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} = 0 \] Here, \(T\) represents the temperature distribution function. The Laplace equation signifies that there is no internal source or sink of heat within the region of interest. In the context of the semi-infinite plate problem, it ensures that the temperature distribution reaches a steady state, with no net change over time. Applying this equation helps us model the temperature at any point on our plate by balancing the rate of heat flow in and out of that point. This balance leads to a stable, unchanging temperature distribution.

Boundary conditions

In solving physical problems, boundary conditions are crucial as they define the constraints necessary for solutions. For the semi-infinite plate problem, the boundary conditions are:

• The bottom edge of the plate has a temperature given by \( T = f(x) = x \) for \( 0 \leq x \leq 10 \)
• The other three edges (sides and top) are kept at a temperature of \( 0^{\circ} \)

These conditions ensure that the solution to the Laplace equation fits the physical scenario. They help in formulating the problem by defining the environment in which the plate exists. For example, the bottom edge condition \( T = x \) shapes the form and constants in our final solution. Moreover, the zero-temperature conditions on the remaining edges simplify the mathematical process by driving some coefficients to zero.

Separation of variables

Separation of variables is a powerful method for solving partial differential equations (PDEs), such as Laplace’s equation. The method assumes that the solution can be represented as a product of functions, each depending on a single coordinate: \[ T(x, y) = X(x)Y(y) \] Substituting this into the Laplace equation, we get: \[ Y(y) \frac{d^2 X}{dx^2} + X(x) \frac{d^2 Y}{dy^2} = 0 \] Dividing through by \(X(x)Y(y)\) separates the variables, each depending on only one coordinate. This equality implies each side must equal a constant, say \(-k^2\), yielding two ordinary differential equations (ODEs). Solving these ODEs gives us forms for \(X(x)\) and \(Y(y)\):

• \( \frac{d^2 X}{dx^2} + k^2 X = 0 \) with solutions \( A \sin(kx) + B \cos(kx) \)
• \( \frac{d^2 Y}{dy^2} - k^2 Y = 0 \) with solutions \( C e^{ky} + D e^{-ky} \)

Separation of variables transforms the PDE into simpler ODEs, which are easier to solve. This method provides a structured path to apply boundary conditions and find specific solutions.

Fourier series

Fourier series are used to express a function as a sum of sinusoids. In the context of the semi-infinite plate problem, we utilize Fourier series to match the boundary condition at the bottom edge. The temperature distribution function at the bottom edge is: \[ T(x, 0) = x = \sum_{n=1}^{\infty} a_n \sin \left( \frac{n \pi x}{10} \right) \], where \( a_n \) are the Fourier coefficients given by: \[ a_n = \frac{2}{10} \int_0^{10} x \sin \left( \frac{n \pi x}{10} \right) dx = \frac{40}{(n \pi)^2} (1 - (-1)^n) \] For odd terms only, these coefficients simplify the solution. Fourier series allow us to represent the complex boundary condition as a sum of simpler sinusoidal modes. Combining these modes helps us to describe the entire steady-state temperature distribution. Ultimately, this leads to the final solution: \[ T(x, y) = \sum_{n=1,3,5,\ldots} \frac{80}{(n \pi)^2} \sin \left( \frac{n \pi x}{10} \right) e^{- \frac{n \pi y}{10}} \] This form leverages Fourier series to match the initial condition and helps us understand the steady-state behavior of the temperature on the plate.