Question

A crossflow heat exchanger with both fluids unmixed has an overall heat transfer coefficient of \(200 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), and a heat transfer surface area of \(400 \mathrm{~m}^{2}\). The hot fluid has a heat capacity of \(40,000 \mathrm{~W} / \mathrm{K}\), while the cold fluid has a heat capacity of \(80,000 \mathrm{~W} / \mathrm{K}\). If the inlet temperatures of both hot and cold fluids are \(80^{\circ} \mathrm{C}\) and $20^{\circ} \mathrm{C}\(, respectively, determine \)(a)$ the exit temperature of the hot fluid and \((b)\) the rate of heat transfer in the heat exchanger.

Short answer

Answer: The exit temperature of the hot fluid is approximately 31.1°C, and the rate of heat transfer in the heat exchanger is approximately 1,956,000 W.

Step-by-step solution

Calculate the heat capacity rates of hot and cold fluids

First, calculate the heat capacity rates (C) of the hot and cold fluids using the following formula: \(C=\dot{m} \times C_{p}\) For the hot fluid, we have: \(C_{h} = 40,000 \frac{\mathrm{W}}{\mathrm{K}}\) And for the cold fluid, we have: \(C_{c} = 80,000 \frac{\mathrm{W}}{\mathrm{K}}\)

Determine the minimum heat capacity rate (Cmin)

In order to use the effectiveness-NTU method, we need to find the minimum heat capacity rate (Cmin) which is the smaller of C_h and C_c: \(C_{\min }=\min \left(C_{h}, C_{c}\right)\) Here, \(C_h = 40,000 W/K\) and \(C_c = 80,000 W/K\). So, the minimum heat capacity rate is: \(C_{\min }=C_{h}=40,000 \frac{\mathrm{W}}{\mathrm{K}}\)

Calculate the NTU (Number of Transfer Units) of the heat exchanger

Next, we calculate the NTU (Number of Transfer Units) by using the overall heat transfer coefficient (U), heat transfer surface area (A), and Cmin from step 2: \(NTU=\frac{UA}{C_{\min }}\) Plug in the given values: \(NTU=\frac{200 \frac{\mathrm{W}}{\mathrm{m}^{2} \mathrm{K}} \times 400 \mathrm{m}^{2}}{40,000 \frac{\mathrm{W}}{\mathrm{K}}} = 2\)

Determine the effectiveness of the heat exchanger

In crossflow heat exchangers with both fluids unmixed, the effectiveness (ε) can be determined using the following approximate formula: \(\epsilon \approx 1-e ^{-NTU}\) Plug in the value of NTU from step 3: \(\epsilon \approx 1-e ^{-2} \approx 0.865\)

Calculate the exit temperature of the hot fluid

Using the effectiveness calculated in step 4, we can find the exit temperature (T_h_out) of the hot fluid using the following formula: \(T_{h_{out}}=T_{h_{in}}-\epsilon \Delta T_{in}\) Here, \(T_{h_{in}}\) is the inlet temperature of the hot fluid (80°C) and \(\Delta T_{in}\) is the temperature difference between the hot and cold fluids at the inlet (80°C - 20°C = 60°C). Plug in the values: \(T_{h_{out}}=80^{\circ} \mathrm{C}-0.865 \times 60^{\circ} \mathrm{C} \approx 31.1^{\circ} \mathrm{C}\) The exit temperature of the hot fluid is approximately 31.1°C.

Calculate the rate of heat transfer in the heat exchanger

The rate of heat transfer (Q) can be determined by using the minimum heat capacity rate (Cmin) and the temperature difference of the hot fluid: \(Q=C_{\min }\left(T_{h_{in}}-T_{h_{out}}\right)\) Plug in the values: \(Q=40,000 \frac{\mathrm{W}}{\mathrm{K}}\left(80^{\circ} \mathrm{C}-31.1^{\circ} \mathrm{C}\right) \approx 1,956,000 \mathrm{~W}\) The rate of heat transfer in the heat exchanger is approximately 1,956,000 W. In conclusion, the exit temperature of the hot fluid is approximately 31.1°C, and the rate of heat transfer in the heat exchanger is approximately 1,956,000 W.