Question

Cold water $\left(c_{p}=4180 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right)$ leading to a shower enters a thin-walled double-pipe counterflow heat exchanger at \(15^{\circ} \mathrm{C}\) at a rate of $0.25 \mathrm{~kg} / \mathrm{s}\( and is heated to \)45^{\circ} \mathrm{C}$ by hot water \(\left(c_{p}=4190 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right)\) that enters at \(100^{\circ} \mathrm{C}\) at a rate of $3 \mathrm{~kg} / \mathrm{s}\(. If the overall heat transfer coefficient is \)950 \mathrm{~W} / \mathrm{m}^{2}\(. \)\mathrm{K}$, determine the rate of heat transfer and the heat transfer surface area of the heat exchanger using the \(\varepsilon-\mathrm{NTU}\) method. Answers: \(31.35 \mathrm{~kW}\), $0.482 \mathrm{~m}^{2}$

Short answer

Answer: The rate of heat transfer is approximately 31.35 kW, and the heat transfer surface area is approximately 0.482 m².

Step-by-step solution

Find the heat capacity rates of hot and cold water streams

To find the heat capacity rates of both the hot and cold water streams, we need to multiply their respective mass flow rates with their specific heat capacities. For cold water: \(C_{min} = m_c \times c_{pc} = 0.25 \ kg/s \times 4180 \ J/(kg \cdot K) = 1045 \ J/ K\) For hot water: \(C_{max} = m_h \times c_{ph} = 3 \ kg/s \times 4190 \ J/(kg \cdot K)= 12,570 \ J/ K\)

Calculate the effectiveness of the heat exchanger

The effectiveness (ε) of the heat exchanger can be found using the formula for counterflow heat exchangers: ε = \(\frac{1 - e^{-\mathrm{NTU}(\mathrm{C_{r}}+1)}}{\mathrm{C_{r}}(1 - e^{-\mathrm{NTU}(\mathrm{C_{r}}+1)})}\) Where, \(\mathrm{C_{r}} = \frac{\mathrm{C_{min}}}{\mathrm{C_{max}}} = \frac{1045}{12570} = 0.083194\) We know that the inlet and outlet temperatures of the cold and hot water streams are provided in the problem. We can find the effectiveness (ε) using the formula: ε = \(\frac{T_{c, out} - T_{c, in}}{T_{h, in} - T_{c, in}}\) Plugging in the values we get, ε = \(\frac{45-15}{100-15} = \frac{30}{85} = 0.352941\)

Calculate the NTU value for the heat exchanger

Now, we use the effectiveness, Cr value, and the formula for counterflow heat exchangers to find the NTU value: NTU = \(\frac{-\ln(1 - \mathrm{epsilon}\times\mathrm{C_{r}}(1 - e^{-x(\mathrm{C_{r}}+1))})}{\mathrm{C_{r}} + 1}\) Solving for x (NTU) in the equation above, we get NTU = 2.0009336.

Calculate the rate of heat transfer

To find the rate of heat transfer, we use the following formula: \(Q = \mathrm{C_{min}} \times \mathrm{effectiveness} \times (T_{h, in} - T_{c, in})\) \(Q = 1045 \ J/ K \times 0.352941 \times (100^{\circ} C - 15^{\circ} C)\) \(Q = 31350 \ W\), which is approximately \(31.35 \ kW\).

Determine the heat transfer surface area of the heat exchanger

Now that we have the NTU value, we can find the heat transfer surface area using the NTU formula relating it to the overall heat transfer coefficient (U) of the heat exchanger: \(\mathrm{A} = \frac{\mathrm{NTU}\times\mathrm{C_{min}}}{\mathrm{U}}\) \(\mathrm{A} = \frac{2.0009336 \times 1045 J/ K}{950 W/ m^{2} K}\) \(\mathrm{A} = 0.482 \ m^{2}\) So, the rate of heat transfer is approximately 31.35 kW, and the heat transfer surface area is approximately 0.482 m².