Chapter 17, Problem 17.37P

Interpretation Introduction

Interpretation:

The allowed range of diameter for copper wire should be determined.

Concept introduction:

Thermal conductivity parallel to fiber is defined by the relation,

Relation for modulus of elasticity is given as:

  $k_C=f_{Cu}{k}_{Cu}+f_{Ag}{k}_{Ag}$

  $f_{Cu}{k}_{Cu}$ are the volume fraction and thermal conductivity for copper.

  $f_{Ag}{k}_{Ag}$ are the volume fraction and thermal conductivity for silver.

Volume fraction is defined as:

  $\begin{array}{l}f=\frac{V_1}{V_1+V_2}\\ f=\text{volume}\text{fraction}\\ V_1=\text{volume}\text{of}\text{first}\text{component}\\ V_2=\text{volume}\text{of}\text{second}\text{component}\\ \text{Where}{\text{V}}_{\text{1}}\text{and}{\text{V}}_{\text{2}}\\ {\text{V}}_{\text{1}}\text{=}\frac{\text{weight}\text{of}\text{first}\text{component}}{\text{density}\text{of}\text{first}\text{component}}\\ {\text{V}}_{\text{2}}\text{=}\frac{\text{weight}\text{of}\text{second}\text{component}}{\text{density}\text{of}\text{second}\text{component}}\\ \text{in}\text{terms}\text{of}\text{weight}\text{and}\text{density}\end{array}$

  $f=\frac{{\text{V}}_{\text{1}}\text{=}\frac{\text{weight}\text{of}\text{first}\text{component}}{\text{density}\text{of}\text{first}\text{component}}}{{\text{V}}_{\text{1}}\text{=}\frac{\text{weight}\text{of}\text{first}\text{component}}{\text{density}\text{of}\text{first}\text{component}}+V_2=\frac{\text{weight}\text{of}\text{second}\text{component}}{\text{density}\text{of}\text{second}\text{component}}}$

The above relation is used for calculation of volume fractions and density of composite

A rule of mixtures for particulate composites is:

  $\begin{array}{l}{\rho }_c={\displaystyle \sum \left(f_i{\rho }_i\right)}\text{for}\left\{\text{i}\text{=}\text{1}\mathrm{..........}\text{n}\right\}\\ {\rho }_c=f_1{\rho }_1+f_2{\rho }_2\mathrm{...................}{f}_n{\rho }_n\end{array}$

Where

  ${\rho }_c=$ density of composite

  $f_i=$ volume fraction of i component

  ${\rho }_i=$ density of i component