Consider an oil-to-oil double-pipe heat exchanger whose flow arrangement is not known. The temperature measurements indicate that the cold oil enters at \(20^{\circ} \mathrm{C}\) and leaves at \(55^{\circ} \mathrm{C}\), while the hot oil enters at \(80^{\circ} \mathrm{C}\) and leaves at \(45^{\circ} \mathrm{C}\). Do you think this is a parallel-flow or counter-flow heat exchanger? Why? Assuming the mass flow rates of both fluids to be identical, determine the effectiveness of this heat exchanger.

In both parallel-flow and counter-flow heat exchangers, the temperature difference between the hot and cold fluids always allows heat transfer. For a parallel-flow heat exchanger, the hot and cold fluids enter at the same end and flow in the same direction, whereas, for a counter-flow heat exchanger, the hot and cold fluids enter at opposite ends and flow in opposite directions. Given temperature measurements: - Cold oil inlet temperature: \(T_{c, in} = 20^{\circ} \mathrm{C}\) - Cold oil outlet temperature: \(T_{c, out} = 55^{\circ} \mathrm{C}\) - Hot oil inlet temperature: \(T_{h, in} = 80^{\circ} \mathrm{C}\) - Hot oil outlet temperature: \(T_{h, out} = 45^{\circ} \mathrm{C}\) Comparing the temperatures, we can conclude the following: - \(T_{c, out} > T_{h, out}\), so the cold fluid outlet temperature is higher than the hot fluid outlet temperature. - \(T_{h, in} > T_{c, in}\), so the hot fluid inlet temperature is higher than the cold fluid inlet temperature. These temperature differences indicate that heat is being transferred from the hot fluid to the cold fluid, as expected in a heat exchanger. Since the hot fluid is cooling down and the cold fluid is heating up, it implies that the fluids are flowing in opposite directions, which is characteristic of a counter-flow heat exchanger.

Assuming identical mass flow rates for both hot and cold fluids, we can calculate the effectiveness of the heat exchanger using the formula: Effectiveness \(= \dfrac{Q_{actual}}{Q_{max}}\) where \(Q_{actual}\) is the actual heat transfer rate and \(Q_{max}\) is the maximum possible heat transfer rate. To calculate \(Q_{actual}\), we can use the energy balance equation: \(Q_{actual} = m \times c_p \times (T_{h, in} - T_{h, out})\) where \(m\) is the mass flow rate, and \(c_p\) is the specific heat capacity of the fluid. Since the mass flow rates and specific heat capacities are identical for both fluids, this equation simplifies to: \(Q_{actual} = (T_{h, in} - T_{h, out})\) Next, we need to calculate the maximum possible heat transfer rate, \(Q_{max}\). The maximum possible heat transfer occurs when the cold fluid reaches the inlet temperature of the hot fluid. Then, \(Q_{max}\) can be calculated as: \(Q_{max} = (T_{h, in} - T_{c, in})\) Now, we can calculate the effectiveness of the heat exchanger: Effectiveness \(= \dfrac{Q_{actual}}{Q_{max}} = \dfrac{T_{h, in} - T_{h, out}}{T_{h, in} - T_{c, in}}\) Effectiveness \(= \dfrac{80 - 45}{80 - 20} = \dfrac{35}{60} \approx 0.583\) The effectiveness of this counter-flow heat exchanger is approximately 0.583, or 58.3%.