Heated air required for a food-drying process is generated by passing ambient air at \(20^{\circ} \mathrm{C}\) through long, circular tubes \((D=50 \mathrm{~mm}, L=5 \mathrm{~m}\) ) housed in a steam condenser. Saturated steam at atmospheric pressure condenses on the outer surface of the tubes, maintaining a uniform surface temperature of \(100^{\circ} \mathrm{C}\).

(a) If an airflow rate of \(0.01 \mathrm{~kg} / \mathrm{s}\) is maintained in each tube, determine the air outlet temperature \(T_{m, o}\) and the total heat rate \(q\) for the tube.

(b) The air outlet temperature may be controlled by adjusting the tube mass flow rate. Compute and plot \(T_{m, o}\) as a function of \(\dot{m}\) for \(0.005 \leq \dot{m} \leq 0.050\mathrm{~kg} / \mathrm{s}\). If a particular drying process requires approximately \(1 \mathrm{~kg} / \mathrm{s}\) of air at \(75^{\circ} \mathrm{C}\), what design and operating conditions should be prescribed for the air heater, subject to the constraint that the tube diameter and length be fixed at \(50 \mathrm{~mm}\) and \(5 \mathrm{~m}\), respectively?