Finite Difference method to solve diffusion equation

In summary, the conversation discusses the application of the finite difference method to solve the transient conduction of a material with given properties and boundary conditions. The Crank-Nicholson method is suggested as a solution, but the individual is unsure how to apply it. The possibility of using the FTCS scheme is also mentioned. Finally, the issue of starting temperature is raised as the given boundary conditions do not provide a starting temperature for comparison.
  • #1
miniradman
196
0

Homework Statement


Plot the transient conduction of a material with k = 210 w/m K, Cp = 350 J/kg K, ρ = 6530 kg/m3

Where the material is a cylinder, with constant cross sectional area and is well insulated. The boundary conditions for the cylinder:
T(0,t) = 330K
T(l,t) = 299K
[itex]\frac{∂T}{∂t}(x,17)=0[/itex]

Homework Equations


Diffusion equation:
[itex]\frac{∂T}{∂t}=\alpha\frac{∂^{2}T}{∂x^{2}}[/itex]

rearranged diffusion equation in finite difference form
[itex]T(x,t+Δt)=\frac{\alphaΔt}{Δx^{2}}[T(x+Δx, t)-2T(x,t)+T(x-Δx,t)][/itex]


The Attempt at a Solution


Hi all

I've never used the finite difference equation before to solve a PDE and I'm unsure how to use it. I know how to find values such as α (thermal diffusivity), but I'm unsure on how to sub in my initial and boundary conditions. And which values would I use for T(x+Δx, t) or T(x-Δx,t)? Since I'm trying to find change in temperature over time at a fixed distance x, I would assume that Δx = 0? (which I know is incorrect).

I've tried looking online for PDE finite analysis techniques, but they're all either ODE examples or mesh analysis (something we haven't covered).
 
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  • #2
Try Googling the Crank-Nicholson method for the 1D heat conduction equation (or the diffusion equation). This is a standard example in courses on finite difference, numerical method, and PDEs.

I don't understand your boundary condition ##\frac { \partial T} {\partial t}(x,17) = 0##. Why t= 17? If the time derivative is constant everywhere at one time, isn't it constant everywhere at all times - i.e. the solution is steady state heat conduction along a uniform rod with the end temperatures given, so there is nothing to solve for numerically!
 
  • #3
Sorry I made a mistake with that, I was meant to say that the system reaches steady state at approximately t = 17.

We haven't yet covered the Crank-Nicholson method, so I have a feeling we may have to stick with the explicit method.
 
  • #4
If you want an explicit method you could use http://en.wikipedia.org/wiki/FTCS_scheme

With the constant temperatures at T(0,t) and T(l,t) the solution will converge to steady state as t increases for any starting temperature distribution T(x,0). It seems a bit strange that the question doesn't specify the starting temperature, so you can have something to check your results against. Just saying that it approximately converges by t = 17 doesn't tell you much.
 
  • #5


I understand your confusion and uncertainty with using the finite difference method to solve a diffusion equation. It can be a complex and unfamiliar approach for those who are new to it. However, I can provide some guidance to help you understand and apply the method to this specific problem.

Firstly, the diffusion equation is a partial differential equation (PDE) that describes the behavior of a diffusive process, such as heat transfer. In this case, we are interested in solving for the temperature distribution in a cylindrical material over time.

The finite difference method is a numerical approach that discretizes the continuous PDE into a set of algebraic equations, making it easier to solve. In this method, the domain of the PDE is divided into a grid of discrete points, with a fixed distance Δx between each point. The values of the dependent variable (temperature in this case) at these points are then approximated using the values at neighboring points.

To apply the finite difference method to this problem, we first need to define the grid and determine the values of Δx and Δt. Since the material is a cylinder with constant cross-sectional area, we can use a one-dimensional grid with Δx representing the distance along the length of the cylinder. Δt represents the time step, which can be chosen based on the desired accuracy and stability of the solution.

Next, we need to discretize the diffusion equation using the finite difference formula you have provided. This will give us an expression for the temperature at each point on the grid at a specific time step. We can then use this formula to iteratively solve for the temperature at each point at successive time steps, starting from the initial conditions.

To incorporate the boundary conditions, we simply apply them at the corresponding points on the grid. For example, at the boundary x=0, we know that the temperature is fixed at 330K, so we can simply set T(0,t)=330K for all time steps. Similarly, at the boundary x=l, we have T(l,t)=299K. The condition \frac{∂T}{∂t}(x,17)=0 represents the fact that the change in temperature at a fixed distance x is zero at t=17, which can be incorporated into the finite difference formula.

Overall, the finite difference method is a powerful tool for solving PDEs, but it does require some understanding and practice to apply it correctly. I would suggest looking for more resources
 

1. What is the Finite Difference method used for?

The Finite Difference method is a numerical technique commonly used to solve differential equations, specifically the diffusion equation. It involves dividing the continuous domain into a discrete grid and approximating the derivatives at each grid point using finite difference approximations.

2. How does the Finite Difference method work?

The Finite Difference method works by discretizing the continuous domain into a grid of equally spaced points. Then, the derivatives at each grid point are approximated using finite difference formulas. These approximations are then substituted into the differential equation, resulting in a system of algebraic equations that can be solved to obtain a numerical solution.

3. What are the advantages of using the Finite Difference method?

One of the main advantages of the Finite Difference method is its simplicity and ease of implementation. It also has a relatively low computational cost compared to other numerical methods. Additionally, it can be applied to a wide range of differential equations and is particularly effective for solving problems with simple geometries.

4. What are the limitations of the Finite Difference method?

The Finite Difference method is limited by the accuracy of the finite difference approximations used. This can result in errors, especially when the grid spacing is large or when the solution has steep gradients. It is also not suitable for problems with complex geometries or boundary conditions.

5. How is the stability of the Finite Difference method determined?

The stability of the Finite Difference method is determined by the Courant-Friedrichs-Lewy (CFL) condition, which states that the time step used in the method must be small enough to ensure that the solution does not become unstable. This condition depends on the grid spacing and the diffusion coefficient of the problem and can be calculated using a stability analysis.

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