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Chapter 11: Problem 10

An air heater in a large coal-fired steam generator heats fresh air entering the steam generator by cooling flue gas leaving the steam generator. One million lbm/hr of air at \(100^{\circ} \mathrm{F}\) and 1.1 million lbm/hr of flue gas at \(720^{\circ} \mathrm{F}\) enters the air heater. The flue gas leaves at \(310^{\circ} \mathrm{F}\). Flue gas has \(c_{p}=0.26 \mathrm{Btu} / \mathrm{lbm}^{\circ} \mathrm{F}\) and \(k=1.39 .\) Pressure changes are small and may be neglected. Calculate the temperature of the air leaving the air heater and the total entropy change for the process.

Short Answer

Expert verified
Final temperature of the air leaving the air heater and the total entropy change for the process are yet to be calculated in the steps described above. Solve these equations at each step carefully to derive these values.

Step by step solution

01

Calculate Heat Loss by Flue Gas

First, calculate the heat loss by the flue gas using the formula \( Q=\dot{m}_{\text{fg}} \cdot c_{p_{\text{fg}}} \cdot (T_{\text{in}_{\text{fg}}} - T_{\text{out}_{\text{fg}}})\), where \( \dot{m}_{\text{fg}} \) is the mass flow rate of the flue gas, \( c_{p_{\text{fg}}} \) is the specific heat of the flue gas at constant pressure, \( T_{\text{in}_{\text{fg}}} \) is the initial temperature of the flue gas, and \( T_{\text{out}_{\text{fg}}} \) is the final temperature of the flue gas.
02

Calculate Final Temperature of Air

The heat transferred from the flue gas is gained by the air. Assuming that the heat capacity of the air is constant, the final temperature of the air \( T_{\text{out}_{\text{air}}} \) can be calculated by equating the heat lost by the flue gas to the heat gained by air. Using the formula \( T_{\text{out}_{\text{air}}} = T_{\text{in}_{\text{air}}} + \frac{Q}{\dot{m}_{\text{air}} \cdot c_{\text{p_{\text{air}}}} \), where \( T_{\text{in}_{\text{air}}} \) is the initial air temperature, \( \dot{m}_{\text{air}} \) is the air mass flow rate and \( c_{\text{p_{\text{air}}} \) is the air specific heat capacity at constant pressure.
03

Calculate the Entropy Change of the Flue Gas

Once the heat transferred is known, calculate the entropy change of the flue gas using the formula \( \Delta S_{\text{fg}} = \int_{T_{\text{in}_{\text{fg}}}}^{T_{\text{out}_{\text{fg}}}} \frac{\dot{m}_{\text{fg}} \cdot c_{\text{p_{\text{fg}}}}}{T}dT \) .
04

Calculate the Entropy Change of the Air

Calculate the entropy change of the air following the equation \( \Delta S_{\text{air}} = \int_{T_{\text{in}_{\text{air}}}}^{T_{\text{out}_{\text{air}}}} \frac{\dot{m}_{\text{air}} \cdot c_{\text{p_{\text{air}}}}}{T}dT \).
05

Calculate Total Entropy Change

The total entropy change for the process is the sum of the entropy changes of the flue gas and air, i.e., \( \Delta S_{\text{total}} = \Delta S_{\text{fg}} + \Delta S_{\text{air}} \).

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