A concentric tube heat exchanger for cooling lubricating oil consists of a thin-walled inner tube of \(25-\mathrm{mm}\) diameter carrying water and an outer tube of \(45-\mathrm{mm}\) diameter carrying the oil. The exchanger operates in counterflow with an overall heat transfer coefficient of \(55 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) and the tabulated average properties.

(a) If the outlet temperature of the oil is \(50^{\circ} \mathrm{C}\), determine the total rate of heat transfer and the outlet temperature of the water.

(b) Determine the length required for the heat exchanger.

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Water m 0.1 kg/s 1 To,out Properties p (kg/m) C, (J/kg -K) Twain = 30C L Water 1000 4200 v (m/s) 7 x 107 k (W/m - K) 0.64 Pr 4.7 Ton=100C Oil m = 0.1 kg/s w.out Oil 800 1900 1 x 10-5 0.134 140