Hot exhaust gases of a stationary diesel engine are to be used to generate steam in an evaporator. Exhaust gases \(\left(c_{p}=1051 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right)\) enter the heat exchanger at \(550^{\circ} \mathrm{C}\) at a rate of \(0.25 \mathrm{~kg} / \mathrm{s}\) while water enters as saturated liquid and evaporates at \(200^{\circ} \mathrm{C}\left(h_{f g}=1941 \mathrm{~kJ} / \mathrm{kg}\right)\). The heat transfer surface area of the heat exchanger based on water side is \(0.5 \mathrm{~m}^{2}\) and overall heat transfer coefficient is \(1780 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Determine the rate of heat transfer, the exit temperature of exhaust gases, and the rate of evaporation of water.

The heat transfer equation is given by: \(q = UA(T_{in} - T_{out})\), where \(q\) is the rate of heat transfer, \(U\) is the overall heat transfer coefficient, \(A\) is the heat transfer surface area, \(T_{in}\) is the initial temperature and \(T_{out}\) is the final temperature. We have the values for \(U\), \(A\), and \(T_{in}\). We need to find \(q\) and \(T_{out}\). First, let's find \(q\). We can rewrite the heat transfer equation as: \(q = mA(T_{in} - T_{out})=m_water \times h_{fg}\), where \(m\) is the mass flow rate of the exhaust gas, \(A\) is the specific heat capacity of exhaust gas, \(T_{in}\) is the initial temperature of exhaust gas, \(T_{out}\) is the final temperature of exhaust gas, and \(m_water\) and \(h_{fg}\) are (respectively) the mass flow rate of water and the specific enthalpy of water. Rearrange the equation to solve for \(q\): \(q = \frac{m_water \times h_{fg}}{A}\).

Solve the equation we found in step 1 for \(m_water\): \(m_water = \frac{q}{h_{fg}}\). Now we can use the heat transfer equation to find the change in temperature of the exhaust gases: \(\Delta T = T_{in} - T_{out} = \frac{q}{mc_p}\).

To find the final temperature of the exhaust gases (\(T_{out}\)), we can subtract the change in temperature (\(\Delta T\)) found in step 2 from the initial temperature (\(T_{in}\)): \(T_{out} = T_{in} - \Delta T\).

Now, we can calculate the rate of evaporation of water by using the equation from step 1 and the heat transfer equation we used in step 2: \(Rate\ of\ evaporation = \frac{m_water}{\Delta T} = \frac{q}{h_{fg}\Delta T}\).