Question

A counterflow double-pipe heat exchanger with \(A_{s}=9.0 \mathrm{~m}^{2}\) is used for cooling a liquid stream \(\left(c_{p}=3.15\right.\) $\mathrm{kJ} / \mathrm{kg} \cdot \mathrm{K})\( at a rate of \)10.0 \mathrm{~kg} / \mathrm{s}$ with an inlet temperature of \(90^{\circ} \mathrm{C}\). The coolant \(\left(c_{p}=4.2 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}\right)\) enters the heat exchanger at a rate of \(8.0 \mathrm{~kg} / \mathrm{s}\) with an inlet temperature of \(10^{\circ} \mathrm{C}\). The plant data gave the following equation for the overall heat transfer coefficient in $\mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}: U=600 /\left(1 / \dot{m}_{c}^{0.8}+2 / \dot{m}_{\mathrm{h}}^{0.8}\right)\(, where \)\dot{m}_{c}\( and \)\dot{m}_{k}$ are the cold- and hot-stream flow rates in \(\mathrm{kg} / \mathrm{s}\), respectively. ( \(a\) ) Calculate the rate of heat transfer and the outlet stream temperatures for this unit. (b) The existing unit is to be replaced. A vendor is offering a very attractive discount on two identical heat exchangers that are presently stocked in its warehouse, each with $A_{s}=5 \mathrm{~m}^{2}$. Because the tube diameters in the existing and new units are the same, the preceding heat transfer coefficient equation is expected to be valid for the new units as well. The vendor is proposing that the two new units could be operated in parallel, such that each unit would process exactly one-half the flow rate of each of the hot and cold streams in a counterflow manner, hence, they together would meet (or exceed) the present plant heat duty. Give your recommendation, with supporting calculations, on this replacement proposal.

Short answer

Provide the necessary information to calculate the outlet stream temperatures, heat transfer rates, and the effectiveness of the existing unit and the proposed parallel units.

Step-by-step solution

Part (a) - Outlet stream temperatures and heat transfer rate calculation

1. Calculate the heat capacity rates for both the hot and cold streams. \(c_{\text{hot}} = m_{h} \cdot c_{p\text{, hot}}\) \(c_{\text{cold}} = m_{c} \cdot c_{p\text{, cold}}\) 2. Calculate the effectiveness, \(ε\), of the heat exchanger using the \(NTU\) method: \(ε = \frac{1 - e^{-NTU(1 - R)}}{1 - R\ e^{-NTU(1 - R)}}\), where \(NTU = \frac{U \cdot A_s}{C_\text{min}}\) \(R = \frac{C_\text{min}}{C_\text{max}}\) 3. Calculate the heat transfer rate, \(Q\), using the effectiveness and the heat capacity rates: \(Q = ε \cdot C_\text{min} \cdot (T_{h\text{, in}} - T_{c\text{, in}})\) 4. Calculate the outlet temperatures of both hot and cold streams: \(T_{h\text{, out}} = T_{h\text{, in}} - \frac{Q}{c_\text{hot}}\) \(T_{c\text{, out}} = T_{c\text{, in}} + \frac{Q}{c_\text{cold}}\)

Part (b) - Evaluating the proposed replacement of two parallel heat exchangers

1. Calculate the new heat capacity rates for both the hot and cold streams for each parallel heat exchanger unit, considering that each unit handles half of the flow rate: \(c_{\text{hot-new}} = \frac{m_{h}}{2} \cdot c_{p\text{, hot}}\) \(c_{\text{cold-new}} = \frac{m_{c}}{2} \cdot c_{p\text{, cold}}\) 2. Calculate the new effectiveness, \(ε_\text{new}\), of each parallel heat exchanger using the \(NTU\) method with the new area, \(A_{s\text{, new}} = 5\ \text{m}^2\): \(ε_\text{new} = \frac{1 - e^{-NTU_\text{new}(1 - R_\text{new})}}{1 - R_\text{new}\ e^{-NTU_\text{new}(1 - R_\text{new})}}\), where \(NTU_\text{new} = \frac{U \cdot A_{s\text{, new}}}{C_\text{min-new}}\) \(R_\text{new} = \frac{C_\text{min-new}}{C_\text{max-new}}\) 3. Calculate the combined heat transfer rate, \(Q_\text{new-total}\), for the two parallel heat exchangers using the new effectiveness and the new heat capacity rates: \(Q_\text{new-total} = 2 \cdot ε_\text{new} \cdot C_\text{min-new} \cdot (T_{h\text{, in}} - T_{c\text{, in}})\) 4. Compare the total heat transfer rates of the existing unit and the proposed parallel units. If \(Q_\text{new-total}\) is greater than or equal to the existing heat transfer rate, Q, then the proposed replacement heat exchangers will meet or exceed the present plant heat duty. Otherwise, the proposal is not recommended.