Question

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

Short answer

Solve the heat equation and apply boundary conditions to find the steady-state temperature distribution.

Step-by-step solution

Identify Boundary Conditions

The boundary conditions for the problem are given as follows:1. The temperature is 0 along the two long sides: \[ T(x, 0) = 0 \] and \[ T(x, 40) = 0 \]2. The temperature is 0 along one short end: \[ T(0, y) = 0 \]3. The temperature distribution along the other short end (x-axis) is: \[ T(x, 0) = \begin{cases} 100^{\circ} & \text{for } 0 < x < 10 \ 0^{\circ} & \text{for } 10 < x < 30 \end{cases} \]

Formulate the Heat Equation

Since we are dealing with steady-state heat distribution, we use Laplace's equation in two dimensions: \[ \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} = 0 \]

Separation of Variables

Assume the solution can be written in the form of a product of two functions: \[ T(x, y) = X(x) \cdot Y(y) \]Substituting into the Laplace equation gives us: \[ X''(x)Y(y) + X(x)Y''(y) = 0 \]Dividing by \(X(x)Y(y)\) results in: \[ \frac{X''(x)}{X(x)} = -\frac{Y''(y)}{Y(y)} = \lambda \]where \(\lambda\) is a separation constant.

Solve the Resulting ODEs

We now have two ordinary differential equations (ODEs):\[ X'' + \lambda X = 0 \]\[ Y'' - \lambda Y = 0 \]The solutions to these ODEs depend on the sign of \(\lambda\). By inspecting the boundary conditions, we will use \( \lambda = \left( \frac{n \pi}{30} \right)^2 \) to ensure non-trivial solutions: \[ X_n(x) = A_n \sin\left( \frac{n \pi x}{30} \right) \]\[ Y_n(y) = B_n \sinh\left( \frac{n \pi y}{30} \right) \]

Form the General Solution

The general solution for \(T(x, y)\) becomes a sum of these solutions for different values of \(n\):\[ T(x, y) = \sum_{n=1}^{\infty} \left[ A_n \sin\left( \frac{n \pi x}{30} \right) \sinh\left( \frac{n \pi y}{30} \right) \right] \]

Apply Initial Conditions

To satisfy the initial condition at \(y = 0\), we use a Fourier series:\[ T(x, 0) = \begin{cases} 100^{\circ} & \text{for } 0 < x < 10 \ 0^{\circ} & \text{for } 10 < x < 30 \end{cases} \]This gives:\[ 100 = \sum_{n=1}^{\infty} A_n \sin\left( \frac{n \pi x}{30} \right) \]for \(0 < x < 10\) and \(A_n = 0\) for \(10 < x < 30\). Solve for \(A_n\) by multiplying both sides by \( \sin \left( \frac{m \pi x}{30} \right) \) and integrating over \(x\): \[ A_n = \frac{2}{30} \int_0^{10} 100 \sin\left( \frac{n \pi x}{30} \right) dx \]

Key concepts

heat equation

The heat equation is crucial in understanding how temperature distributes over time within a material. For steady-state problems, the temperature does not change over time. The mathematical form simplifies to Laplace's equation. For a plate, the steady-state condition is given by \[ \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} = 0 \] This equation ensures that the sum of the second derivatives of temperature with respect to spatial coordinates is zero. It describes how heat spreads out evenly until a stable temperature is reached throughout the material.

boundary conditions

Boundary conditions are essential to define a unique solution for the heat equation. For a rectangular plate with steady-state temperature distribution, boundary conditions might include:

• The temperature is fixed along certain edges.
• The temperature can vary in specific ways along other edges.

In our problem:

• Temperature is 0 along the two long sides:
\[ T(x, 0) = 0 \] \[ T(x, 40) = 0 \]
• Temperature is 0 along one short end:
\[ T(0, y) = 0 \]
• Temperature distribution along the other short end (x-axis) is:
\[ T(x, 0) = \begin{cases} 100^{\circ} & \text{for } 0 < x < 10 \ 0^{\circ} & \text{for } 10 < x < 30 \end{cases} \] These conditions guide how the heat spreads within the plate to ensure the unique temperature distribution.

separation of variables

Separation of variables is a powerful technique to solve partial differential equations like the heat equation. We assume that the temperature function can be written as a product of two functions, one dependent on x and the other on y: \[ T(x, y) = X(x) \cdot Y(y) \] Substituting this into Laplace's equation results in: \[ X''(x)Y(y) + X(x)Y''(y) = 0 \] Dividing both sides by \(X(x)Y(y)\) isolates the variables: \[ \frac{X''(x)}{X(x)} = -\frac{Y''(y)}{Y(y)} = \lambda \] Here, \(\lambda\) is a separation constant, and it leads to two ordinary differential equations (ODEs), one in x and one in y. This simplifies solving the problem by breaking it down into more manageable parts.

Laplace's equation

Laplace's equation is fundamental for steady-state heat distribution problems. It states that the second partial derivatives of the temperature with respect to spatial coordinates x and y sum to zero: \[ \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} = 0 \] This implies there are no sources or sinks of heat within the plate, so heat spreads evenly until it reaches equilibrium. Solving Laplace's equation with appropriate boundary conditions allows us to find how temperature varies within the material.

Fourier series

Fourier series help represent complex temperature distributions along the boundaries using sums of sines and cosines. In our problem, the boundary condition at y=0 is a piecewise function, which can be expanded into a Fourier series: \[ T(x, 0) = \begin{cases} 100^{\circ} & \text{for } 0 < x < 10 \ 0^{\circ} & \text{for } 10 < x < 30 \end{cases} \] This translates to: \[ 100 = \sum_{n=1}^{\infty} A_n \sin\left( \frac{n \pi x}{30} \right) \] Fourier series simplifies solving the temperature distribution by breaking it down into simpler periodic components.