A 1 -shell and 2-tube type heat exchanger has an overall heat transfer coefficient of \(300 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}^{2} \cdot{ }^{\circ} \mathrm{F}\). The shell side fluid has a heat capacity rate of \(20,000 \mathrm{Btu} / \mathrm{h} \cdot{ }^{\circ} \mathrm{F}\), while the tube side fluid has a heat capacity rate of \(40,000 \mathrm{Btu} / \mathrm{h} \cdot{ }^{\circ} \mathrm{F}\). The inlet temperatures on the shell side and tube side are \(200^{\circ} \mathrm{F}\) and \(90^{\circ} \mathrm{F}\), respectively. If the total heat transfer area is \(100 \mathrm{ft}^{2}\), determine \((a)\) the heat transfer effectiveness and \((b)\) the actual heat transfer rate in the heat exchanger.

We are given the heat capacity rates for the shell side fluid (\(C_{s} = 20,000 \mathrm{Btu} / \mathrm{h} \cdot{ }^{\circ} \mathrm{F}\)) and the tube side fluid (\(C_{t} = 40,000 \mathrm{Btu} / \mathrm{h} \cdot{ }^{\circ} \mathrm{F}\)). The minimum heat capacity rate (\(C_{min}\)) is the smallest of the two heat capacity rates: \(C_{min} = \min(C_{s}, C_{t}) = 20,000 \mathrm{Btu} / \mathrm{h} \cdot{ }^{\circ} \mathrm{F}\) The heat capacity rate ratio (\(C_r\)) is the ratio of the minimum heat capacity rate to the maximum heat capacity rate: \(C_r = \frac{C_{min}}{\max(C_{s}, C_{t})} = \frac{20,000}{40,000} = 0.5\)

The number of transfer units (NTU) is given by the formula: \(NTU = \frac{U \cdot A}{C_{min}}\) where \(U\) is the overall heat transfer coefficient, \(A\) is the total heat transfer area, and \(C_{min}\) is the minimum heat capacity rate. We are given that \(U = 300 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}^{2} \cdot{ }^{\circ} \mathrm{F}\) and \(A = 100 \mathrm{ft}^{2}\). \(NTU = \frac{300 \cdot 100}{20,000} = 1.5\)

For a 1-shell and 2-tube type heat exchanger, the heat transfer effectiveness (\(\epsilon\)) can be calculated using the following formula: \(\epsilon = \frac{1 - e^{-NTU(1+C_r)}(1+C_r)}{1 + C_r}\) Substituting the values of \(NTU\) and \(C_r\): \(\epsilon = \frac{1 - e^{-1.5(1+0.5)}(1+0.5)}{1 + 0.5}\) \(\epsilon \approx 0.662\) So, the heat transfer effectiveness is approximately 0.662.

The actual heat transfer rate (\(q\)) can be determined using the heat transfer effectiveness (\(\epsilon\)), the minimum heat capacity rate (\(C_{min}\)), and the temperature difference between the inlet temperatures of the shell side and tube side fluids (\(T_{s_in} - T_{t_in}\)): \(q = \epsilon \cdot C_{min} \cdot (T_{s_in} - T_{t_in})\) We have already calculated the heat transfer effectiveness (\(\epsilon \approx 0.662\)) and the minimum heat capacity rate (\(C_{min} = 20,000 \mathrm{Btu} / \mathrm{h} \cdot{ }^{\circ} \mathrm{F}\)). The inlet temperatures of the shell side and tube side fluids are \(T_{s_in} = 200^{\circ} \mathrm{F}\) and \(T_{t_in} = 90^{\circ} \mathrm{F}\), respectively. \(q \approx 0.662 \cdot 20,000 \cdot (200 - 90)\) \(q \approx 1,459,280 \mathrm{Btu/h}\) So, the actual heat transfer rate in the heat exchanger is approximately \(1,459,280 \mathrm{Btu/h}\). In conclusion, (a) the heat transfer effectiveness of the heat exchanger is approximately 0.662 and (b) the actual heat transfer rate is approximately \(1,459,280 \mathrm{Btu/h}\).