1D UNSTEADY STATE HEAT CONDUCTION: EXPLICIT METHOD

**This project is a part of an Assignment submitted at Flowthermolab.**

OVERVIEW

1. Problem Statement:
Solve 1D unsteady state heat conduction for a rod of 1m length, and 300^oC base temperature, Temperature at tip, Ttip = 50-degree Celcius and at t=0 sec, Initial Temperature (T_init) is 30^oC
Study the Effect of Grid Size (delx) and Time Step Size (delt)

Unique Work:
Prepared Algorithm, Coded in MATLAB and Verification from the Analytical Calculations.

METHODOLOGY

Tabel 1: Methodology Adopted

Layout	Details
1. Schematic Diagram and Meshing	Figure 1: Diagram specifying the Geometry and Meshing Deatils
2. Defining Governing Equation	Figure 2: Governing Equation for diffusion / heat conduction problem Note:    - No Source term    - No Unsteady term
4. Algorithm	1. Define the geometry: Length (L) [m], density [kg/m^3] 2. Discretize the geometry:   - Define Number of Grids (N)   - Grid size (𝛥𝑥) = Length / Number of grids = L / N   - Define Step size and initialize the time matrix 3. Define Boundary Conditions and Initialize   - Initialize temprature matrix   - Define the values of constants separately for internal and boundary nodes at base and tip 4. Solve for loop to Calculate temperature at nodes 5. Make data visually understandable and clear to first visual users
5. Results: Verification/Validation & Case	Figure 3: Computed Results for fixed time step Note: The values were verified from the textbook ,’ An Introduction to Computational Fluid Dynamics by Versteeg_Malalasekera_2edition. Figure 4: Computed Results for fixed grid size Figure 3: Computed Results for Final Optimal Case

DISCUSSION & CONCLUSION

For Changes in Number of Grids and keeping timestep constant to 1e-3 running for 0.4sec.
     Transition of number of grids at 10, 20, 25, and 30. Graphs were used to represent the data. It can be seen clearly that faster convergence was achieved at 25 number of grids where at 30 number of grid unstable data was recorded.
     Thus, on increasing number of grids faster stability is achieved until a threshold point after which increasing the number of grids will make the system solution unstable. (Similar Observation when number of grids are kept constant)
     Thus, optimal values for both timestep and number of grids were taken to solve the system faster and maintain the stable solution.

• Number of Nodes = 15
• Timestep = 3e-3