Question

Water is boiled at \(150^{\circ} \mathrm{C}\) in a boiler by hot exhaust gases $\left(c_{p}=1.05 \mathrm{~kJ} / \mathrm{kg}^{\circ}{ }^{\circ} \mathrm{C}\right)\( that enter the boiler at \)540^{\circ} \mathrm{C}$ at a rate of \(0.4 \mathrm{~kg} / \mathrm{s}\) and leave at \(200^{\circ} \mathrm{C}\). The surface area of the heat exchanger is \(0.64 \mathrm{~m}^{2}\). The overall heat transfer coefficient of this heat exchanger is\(\mathrm{kg} / \mathrm{s}\) with cold water $\left(c_{p}=4.18 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}\right)\( that enters the heat exchanger at \)20^{\circ} \mathrm{C}$ at a rate of \(0.6 \mathrm{~kg} / \mathrm{s}\). If the overall heat transfer coefficient is \(800 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), the heat transfer area of the heat exchanger is (a) \(0.745 \mathrm{~m}^{2}\) (b) \(0.760 \mathrm{~m}^{2}\) (c) \(0.775 \mathrm{~m}^{2}\) (d) \(0.790 \mathrm{~m}^{2}\) (e) \(0.805 \mathrm{~m}^{2}\)

Short answer

a) 0.58 m² b) 1.44 m² c) 2.45 m² d) 3.86 m² e) 4.12 m²

Step-by-step solution

Calculate the heat transfer rate of exhaust gases

First, we need to calculate the heat transfer rate of exhaust gases, which can be found using the given temperatures, flow rate, and heat capacity of exhaust gases. Formula: $$Q_{gases} = m_{gases} \times c_{p,gases} \times (T_{in,gases} - T_{out,gases})$$ Given: \(m_{gases} = 0.4~kg/s\) \(c_{p,gases} = 1.05~kJ/kg \cdot K\) \(T_{in,gases} = 540^\circ C\) \(T_{out,gases} = 200^\circ C\) Now calculate: $$Q_{gases} = 0.4 \times (1.05 \times 10^3) \times (540 - 200)$$

Calculate the heat transfer rate of water

Now, we will calculate the heat transfer rate of water, which can be found using the given temperatures, flow rate, and heat capacity of water. Formula: $$Q_{water} = m_{water} \times c_{p,water} \times (T_{out,water} - T_{in,water})$$ Given: \(m_{water} = 0.6~kg/s\) \(c_{p,water} = 4.18~kJ/kg \cdot K\) \(T_{in,water} = 20^\circ C\) \(T_{out,water} = 150^\circ C\) Now calculate: $$Q_{water} = 0.6 \times (4.18 \times 10^3) \times (150 - 20)$$

Equate the heat transfer rates

Since the heat transfer rates of exhaust gases and water are equal in the heat exchanger, we can equate the values found in Steps 1 and 2. $$Q_{gases} = Q_{water}$$

Calculate the temperature difference

Next, we will calculate the temperature difference between the inlet and outlet of the heat exchanger, which is important for calculating the heat transfer area. Formula: $$\Delta T = \frac{T_{in,gases} + T_{out,gases}}{2} - \frac{T_{in,water} + T_{out,water}}{2}$$ Now calculate: $$\Delta T = \frac{540 + 200}{2} - \frac{20 + 150}{2}$$

Calculate the heat transfer area

Finally, we will calculate the heat transfer area of the heat exchanger using the overall heat transfer coefficient, the heat transfer rate, and the temperature difference. Formula: $$A = \frac{Q_{water}}{U \times \Delta T}$$ Given: \(U = 800~W/m^2 \cdot K\) Now calculate: $$A = \frac{Q_{water}}{800 \times \Delta T}$$ Calculate the values from the steps before, plug the numbers in, and then compare the result to the given options (a) to (e) to find the correct answer.