Summary of "Exp No 4 CFD analysis of heat transfer in pin fin"

Summary of “Exp No 4 CFD Analysis of Heat Transfer in Pin Fin”

This video presents an experiment focused on the computational fluid dynamics (CFD) analysis of heat transfer in a pin fin, specifically using numerical methods to solve heat transfer equations and understand temperature variation along the fin length. The experiment emphasizes programming and numerical analysis over purely theoretical understanding.

Main Ideas and Concepts

1. Introduction to Heat Transfer in Extended Surfaces (Fins)

Extended surfaces (fins) enhance heat transfer rates.
Heat transfer modes involved:
- Conduction (within the solid fin)
- Convection
- Radiation (between fin surface and surroundings)
Radiation is often neglected unless temperatures exceed approximately 100–150°C.
Different fin shapes include rectangular, triangular, annular, and pin fins with non-uniform cross sections.
Efficiency varies with convection type; natural convection is generally more efficient than forced convection for fins.

2. Governing Heat Transfer Equation

Heat balance on an elemental fin segment leads to a second-order differential equation describing temperature distribution.
The equation accounts for conduction within the fin and convection from the surface.
For fins with variable cross-sectional area, the equation is more complex; for uniform cross-section, it simplifies.
Temperature difference (\Theta = T - T_\infty) is used to simplify the equation.

3. Boundary Conditions

At the fin base ((x=0)): (\Theta = \Theta_B) (base temperature difference).
At the fin tip ((x=L)): insulated tip condition, i.e., zero temperature gradient (\frac{d\Theta}{dx} = 0).

4. Numerical Solution Using Finite Difference Method (FDM)

Analytical solution is difficult due to boundary conditions and equation form.
FDM is used to discretize the differential equation.
Central difference scheme is preferred for its accuracy over forward and backward difference methods.
Discretization involves dividing the fin length into nodes (e.g., 7 nodes for demonstration).
The second derivative is approximated using the central difference formula.
Boundary conditions are converted from differential to algebraic form for numerical solution.

5. Matrix Formulation

The discretized equations form a system of linear equations.
Coefficients for each node are arranged into a coefficient matrix (A).
Known boundary values form the vector (B).
The system (A \Theta = B) is solved to find temperature differences at internal nodes.
Matrix size depends on the number of discretization points (e.g., 5×5 for 7 nodes).

6. Scilab Programming for Numerical Solution

The experiment involves writing Scilab code to:
- Define input parameters: fin diameter, heat transfer coefficient, thermal conductivity for different materials.
- Calculate parameters like (m = \sqrt{\frac{hP}{kA_c}}).
- Set up discretized spatial domain and step size.
- Initialize matrices (A) and (B).
- Populate matrices with coefficients based on finite difference equations.
- Solve the matrix equation using matrix inversion.
- Repeat for three materials: copper, steel, and aluminum (only thermal conductivity (k) changes).
Plot temperature difference (\Theta) vs. fin length (x) for all materials.

7. Results and Analysis

Copper shows the slowest temperature drop along the fin length due to its highest thermal conductivity.
Aluminum performs moderately.
Steel shows the steepest temperature gradient (lowest conductivity).
The plot visually compares temperature distributions, confirming copper’s superior heat transfer performance.

8. Learning Outcomes

Understanding the governing differential equation for heat transfer in fins.
Implementing the central difference finite difference method for numerical solution.
Writing and executing Scilab code to solve matrix equations for temperature distribution.
Interpreting and comparing temperature profiles for different materials.

Methodology / Instructions (Detailed Steps)

Understand the physical problem and governing equations
- Heat conduction and convection in a fin.
- Set boundary conditions.
Discretize the fin length into nodes
- Define number of nodes (n).
- Calculate step size (h = \frac{L}{n}).
Apply central difference approximation
- Approximate second derivative (\frac{d^2 \Theta}{dx^2}) at each node using: [ \frac{\Theta_{i+1} - 2\Theta_i + \Theta_{i-1}}{h^2} ]
Formulate finite difference equations for each internal node
- Incorporate convection term (n^2 \Theta_i).
- Convert boundary differential condition to algebraic form.
Assemble coefficient matrix (A) and vector (B)
- Populate matrix (A) with coefficients from discretized equations.
- Populate vector (B) with known boundary conditions.
Solve the linear system (A \Theta = B)
- Use matrix inversion or built-in solver in Scilab.
Repeat for different materials by changing thermal conductivity (k)
Plot temperature difference vs. fin length
- Compare results for copper, steel, and aluminum.

Speakers / Sources

Primary Speaker: Instructor or lecturer conducting the Mechanical Systems Analysis Laboratory experiment.
No other speakers or external sources explicitly identified.

This summary captures the key theoretical concepts, numerical methodology, programming approach, and analysis presented in the video on CFD heat transfer analysis in pin fins.