$6\mathrm{.}106$ Saturated water vapor at $100$kPa enters a counterflow heat exchanger operating at steady state and exits at $20C$ with a negligible change in pressure. Ambient air at $275K,1$atm enters in a separate stream and exits at $290K,1$ atm. The air mass flow rate is $170$ times that of the water. The air can be modeled as an ideal gas with $c_p=1\mathrm{.}005k\frac{J}{k}g*K.$ Kinetic and potential energy effects can be ignored.

$($a$)$ For a control volume enclosing the heat exchanger, evaluate the rate of heat transfer, in $kJ$ per $kg$ of water flowing.

$($b$)$ For an enlarged control volume that includes the heat exchanger and enough of its immediate surroundings that heat transfer from the control volume occurs at the ambient temperature, $275K,$ determine the rate of entropy production, in $K\frac{J}{K}$ per $kg$ of water flowing.