Problem$-1$:

An electronic device, in the form of a disk $20$ mm in diameter, is mounted flush on a large aluminum alloy

$(2024)$ block whose temperature is maintained at $27C.$ The mounting arrangement is such that a contact

resistance of $R_{t,c}^{\text{'}\text{'}}=5{10}^{-5}{m}^2\frac{K}{W}$ exists at the interface between the device and the block. In order to cool

the electronic device, a thermal engineer proposes to attach a finned heat sink to the top of the device.

The pin fins and base material are fabricated from copper $(k=400\frac{W}{m}K)$ and are exposed to an airstream

at $27C$ for which the convection coefficient is $1000\frac{W}{m^2}K.$ What is the maximum permissible operating

power of the device if the maximum permissible temperature of the device is $70C?$ To improve the cooling

performance, a material scientist develops an epoxy material achieving $5$ times improvement in contact

resistance i$.$e$.R_{t,c}^{\text{'}\text{'}}$ drops down to $1{10}^{-5}{m}^2\frac{K}{W}.$ What is the maximum permissible operating power of

the device after this epoxy is deployed?