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Provide the solution to question.2.75.

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**Solution Number: 2.75**
$$\textbf{Define Mass Flow Variables } \text{The total mass flow rate is constant at } \dot{m}_{total} = 2.5\text{ kg/s. Let } x\text{represent the fraction of hot water, where the hot water mass flow is } \dot{m}_{h} = x \cdot 2.5 \text{ and the}\text{cold water mass flow is } \dot{m}_{c} = (1 - x) \cdot 2.5\text{.}$$
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$$\textbf{Calculate Cross-Sectional Areas } \text{The area for the hot-water line with diameter } D_{h} =0.015\text{ m is } A_{h} = \frac{\pi(0.015)^2}{4} \approx 1.767 \times 10^{-4}\text{ m}^2\text{. The area for the cold-water line with}\text{diameter } D_{c} = 0.02\text{ m is } A_{c} = \frac{\pi(0.02)^2}{4} \approx 3.142 \times 10^{-4}\text{ m}^2\text{.}$$
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$$\textbf{Establish Velocity and Kinetic Energy Functions } \text{Using the provided formula } V = \frac{\dot{m}}{\rho A} \text{ with}\rho = 1000\text{ kg/m}^3\text{, the velocities are } V_{h} = \frac{x \cdot 2.5}{1000 \cdot A_{h}} \text{ and } V_{c} = \frac{(1 - x) \cdot 2.5}{1000 \cdot A_{c}}\text{. The specific kinetic}\text{energy in kJ/kg for each stream is } ke = \frac{V^2}{2000}\text{.}$$
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$$\textbf{Formulate Total Energy Rate Equation } \text{The total rate of energy transfer } \dot{E} \text{ is the sum of the}\text{energy rates from both lines: } \dot{E} = \dot{m}_{h}(h_{h} + ke_{h}) + \dot{m}_{c}(h_{c} + ke_{c})\text{. Substituting the given}\text{enthalpies } h_{h} = 335\text{ kJ/kg and } h_{c} = 21.0\text{ kJ/kg, the equation becomes } \dot{E} =(2.5x)(335 + \frac{V_{h}^2}{2000}) + (2.5(1 - x))(21.0 + \frac{V_{c}^2}{2000})\text{.}$$
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$$\textbf{Analyze the Range for Plotting } \text{As the percentage of hot water varies from 0\% } (x = 0) \text{ to}100\% \text{ } (x = 1)\text{, the energy rate } \dot{E} \text{ will increase significantly. At } x = 0, \dot{E} = 2.5(21.0 +\frac{V_{c}^2}{2000}) \approx 52.5\text{ kW. At } x = 1, \dot{E} = 2.5(335 + \frac{V_{h}^2}{2000}) \approx 837.75\text{ kW (assuming } ke \text{ remains}\text{small relative to } h\text{).}$$

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Problem #18