$($a$)$ A solar$-$powered Stirling engine is designed to operate between a solar collector at $500$ K and ambient air at $300$ K $.$

$($i$)$ Explain why the actual efficiency of the Stirling engine is always less than the Carnot efficiency.

$[2$ marks$]$

$($ii$)$ If the Stirling engine produces $150$ kW of power, and the actual thermal efficiency is $60\%$ of the maximum possible efficiency, calculate the rate of heat input $($in kW $)$ required for the engine.

$[4$ marks$]$

$($b$)$ A data center requires a cooling system to maintain the equipment at $20C$ while rejecting heat to the ambient air at $35C.$ A vapor$-$compression refrigeration cycle is employed, but due to limitations$,$ the coefficient of performance $($COP$)$ achieved is only $70\%$ of the Carnot COP.

$($i$)$ If the cooling system removes heat from the data center at a rate of $200$ kW $,$ calculate the actual power input required to operate the cooling system.

$[5$ marks$]$

$($ii$)$ Evaluate how the performance of the refrigeration system would change if the ambient temperature increased to $40C$ due to a heatwave.

$[4$ marks$]$