Material Selection

Overview of Course

• Title: Properties of Engineering Materials

• Course Code: ES 1021

• Institution: Western Engineering

Material Selection Procedure

• The course material has progressed to a stage where material selection based on performance is feasible.

• There are three vital components necessary for the material selection process:

  • Objective Function: This is a mathematical expression that represents the goal of the design.

  • Constraint Equation: This represents limitations or constraints within the design parameters.

  • Free Variable: This is an element of the design, typically related to geometry, that can be manipulated. It must be included in both the objective and constraint equations.

Example of Material Selection Procedure

• Context: Utilizing the insulating box example previously discussed in the lectures.

• Definitions:

  • Objective: Minimize the mass of the insulating box.

  • Constraint: Heat flux must be less than a maximum value, denoted as $q_{\text{max}}$ .

  • Free Variable: The wall thickness represented as $t$ .

1. Objective Function:

   • The mathematical expression for mass is given by: $mass = A \times t \times \rho$

   • Where:

     • $A$ is the surface area,

     • $t$ is the wall thickness,

     • $\rho$ is the density of the material.

2. Constraint Equation:

   • For heat flux, the relation is: $q<em>{\text{max}} \geq -\frac{\tau(T</em>c - T_h)}{t}$

   • Where:

     • $\tau$ is the thermal conductivity,

     • $T_c$ is the cold side temperature,

     • $T_h$ is the hot side temperature.

3. Rearranging the Constraint:

   • Solve for the free variable $t$:

   • $t \geq -\frac{\tau(T<em>c - T</em>h)}{q_{\text{max}}}$

4. Substituting the Rearranged Constraint into the Objective Function:

   • This yields:

   • $mass \geq -A \times \frac{\tau(T<em>c - T</em>h)}{q_{\text{max}}} \times \rho$

5. Grouping Variables:

   • The variables are categorized into Functional, Geometric, and Material properties:

   • Model: $Obj = F \times G \times M$

   • Resulting in:

   • $mass \geq -\frac{(T<em>c - T</em>h)}{q_{\text{max}}} \times A \times \tau \times \rho$

6. Analysis of Materials:

   • Materials that minimize mass while adhering to constraints will yield the minimal Material Index, represented as: $M = \tau \times \rho$

   • Where:

     • $M$ is the Material Index,

     • $\tau$ is the thermal conductivity,

     • $\rho$ is the density.

Graphical Representation of Material Properties

• A graph plotting $\tau$ vs. $\rho$ on logarithmic axes leads to:

  • $\log M = \log \tau + \log \rho$

  • Rearranged to:

  • $\log \tau = - \log \rho - \log M$

• Implications:

  • Materials with identical values of $M$ will demonstrate equivalent performance characteristics.

  • These materials will align on a line with a slope of -1 on the graph.

  • The most efficient materials will be characterized by the smallest value of $M$, which corresponds to the lowest y-intercept of the graph.

Example Graph Variables

• Density ($\text{kg/m}^3$):

  • 10, 100, 1000, 10000

• Thermal Conductivity ($W/m.\circ C$):

  • 0.01, 0.1, 1, 10, 100, 1000

Additional Material Selection Resources

• Granta EduPack: A resource mentioned for graphical selection demonstration pertaining to material science concepts.

Conclusion

• This segment of the course lays foundational knowledge for material selection based on engineering principles and material properties, emphasizing the importance of understanding objective functions, constraints, and the role of free variables.

Institution

• Western University Canada