Problem $1$: Insulating Walls for Freezers

Objective: Minimize thermal conductivity while maximizing stiffness.

A$.$ Define Function, Constraints, Objective, and Free Variables:

Function:

structural

Provide thermal insulation and stiffness in freezer panels.

Constraints: Bending stiff ness $S^{**}$ specified

Fourier's Law $Q=\frac{*A*T}{\text{t}\text{h}\text{i}\text{c}\text{h}\text{n}\text{e}\text{s}\text{s}}.$ Low thermal conductivity $$ to minimize heat transfer. $$ low link with stiffneso Adequate stiffness to reduce vibrations and allow thinner steel skins. stiffness high

$3.$ Objective:

obrace$($ penalty function ${)}^{\frac{m}{m_0},\frac{}{{}_0}}4.$ Fre

Minimize $$ a $-$ Minimize m

$m=fV$

J

Material of the core $($e$.$g$.,$ polyurethane foam, polystyrene foam, aerogels$).$ thicknes

Core thickness and

B$.$ Material Selection:

Identify relevant material indices such as $$ density ratio or stiffness$/$density ratio.

Use a material selection chart or software to filter potential materials.

Compare $3$ candidate materials $($e$.$g$.,$ polyurethane foam, polystyrene foam, or aerogels$)$ using simulations for stiffness and thermal performance.

Material Indices:

To balance insulation and stiffness:

Thermal insulation index: Low $.$

Stiffness index: High E$/$p $($stiffness$-$to$-$density ratio$).$

Preliminary Material Candidates:

Polyurethane foam:

Low density $(20-100$kg$/$m$3).$

Low $(0\mathrm{.}02\frac{W}{m}K).$

Reasonable stiffness.

we vairate $h($make it min $)//$ HOW,DID WE FIND THE MATERIAL INDEXES AND penalty function.....?????