Heat Equation

Description

The Heat Equation is a MATLAB script that models the 1D heat equation for metal bars made of aluminum (Al), iron (Fe), and copper (Cu). It discretizes space and time to ensure stability and assumes Neumann boundary conditions. The simulation calculates the temperature distribution over time for the metal bars, visualizing the evolution of temperature profiles for each portion from the larger bar.

Physical Parameters

• Characteristics Time: Determines the characteristic time for the system based on the metal bars properties.
• Initial Conditions: Specifies the initial temperatures for each metal bar.
• Diffusion Coefficient: Sets the diffusion coefficient based on the material type.

Dynamic Process

• Temperature Calculation: Utilizes the heat equation and finite difference method to compute the temperature distribution over time.
• Boundary Conditions: Imposes Neumann boundary conditions to regulate heat flux at the boundaries.
• Dynamic Visualization: Provides real-time graphical visualization of the evolving temperature distribution.

Numerical Solution

To solve the heat equation numerically, the finite difference method is employed. This method discretizes both space and time, allowing us to approximate the continuous heat equation using discrete values.

$$ \frac{\partial T}{\partial t} = D \frac{\partial^2 T}{\partial x^2} $$

• where $T$ is the temperature,
• $D$ is the thermal diffusivity,
• $t$ is time, and
• $x$ is position along the metal bar.

The temperature distribution over time is then calculated using a numerical scheme that approximates the heat equation. This scheme typically involves updating the temperature values at each point in space and time based on the neighboring temperature values and the thermal diffusivity of the material.

$$ T_{i+1,j} = T_{i,j} + \frac{{\Delta t}}{{\Delta x^2}} \left( D_{j+1}(T_{i,j+1} - T_{i,j}) - D_{j}(T_{i,j} - T_{i,j-1}) \right) $$

• where $T_{i,j}$ is the temperature at time $i$ and position $j$,
• $D_{j}$ is the thermal diffusivity at position $j$,
• $Delta$ t is the time step, and
• $Delta$ x is the spatial step.

Neumann Boundary Conditions

The Neumann boundary conditions at both ends of the metal bars ensures that the heat flux, or the rate of heat transfer per unit area, is controlled at these boundaries: $$\frac{{\partial T}}{{\partial x}} = 0$$

At both ends of the metal bars, Neumann boundary conditions are enforced. These conditions dictate the heat flux at the boundaries rather than directly setting the temperature. Instead of directly prescribing the temperature values, these conditions govern how heat is allowed to flow into or out of the system through the boundaries.

$$ \frac{{T_{i+1,1} - T_{i,1}}}{{\Delta x}} = 0 \implies T_{i+1,1} = T_{i,1} $$

$$ \frac{{T_{i+1,P} - T_{i,P}}}{{\Delta x}} = 0 \implies T_{i+1,P} = T_{i,P} $$

By specifying the heat flux at the boundaries, we effectively regulate the transfer of thermal energy across them. This approach is particularly useful when modeling scenarios where the physical boundaries are insulated or when the rate of heat transfer is of primary interest.