Fluid with a specific enthalpy of 2 280 ku/kg enters a condenser at the rate of 4 500 kg/h. and leaves with a specific enthalpy of 163 kl/kg If the enthalpy of the cooling water circulating through the condenser tubes increases at the rate of 148 000 kJ/min, determine the rate at which heat energy flows the condenser to the atmosphere.
The correct answer and explanation is :
To determine the rate at which heat energy flows through the condenser to the atmosphere, we will use the principle of energy conservation. Specifically, the heat energy removed from the steam is transferred to the cooling water circulating through the condenser.
Step 1: Understand the given data
- Specific enthalpy of the steam entering the condenser: ( h_{\text{in}} = 2280 \, \text{kJ/kg} )
- Specific enthalpy of the steam leaving the condenser: ( h_{\text{out}} = 163 \, \text{kJ/kg} )
- Mass flow rate of the steam: ( \dot{m}_{\text{steam}} = 4500 \, \text{kg/h} )
- Heat gained by the cooling water: ( Q_{\text{water}} = 148,000 \, \text{kJ/min} )
Step 2: Convert units to a consistent system
First, convert the mass flow rate of the steam from kg/h to kg/min:
[
\dot{m}_{\text{steam}} = \frac{4500 \, \text{kg/h}}{60} = 75 \, \text{kg/min}
]
Step 3: Calculate the heat lost by the steam
The heat removed from the steam is the change in enthalpy multiplied by the mass flow rate:
[
Q_{\text{steam}} = \dot{m}{\text{steam}} \times (h{\text{in}} – h_{\text{out}})
]
Substitute the values:
[
Q_{\text{steam}} = 75 \, \text{kg/min} \times (2280 – 163) \, \text{kJ/kg}
]
[
Q_{\text{steam}} = 75 \times 2117 \, \text{kJ/min} = 158,775 \, \text{kJ/min}
]
Step 4: Analyze the energy balance
The heat removed from the steam is transferred to the cooling water, so the heat energy flowing through the condenser to the atmosphere is equal to the heat gained by the cooling water plus the heat lost by the steam.
Given the rate of heat gain by the cooling water:
[
Q_{\text{water}} = 148,000 \, \text{kJ/min}
]
The heat energy flowing through the condenser to the atmosphere is:
[
Q_{\text{condenser}} = Q_{\text{steam}} = 158,775 \, \text{kJ/min}
]
Step 5: Conclusion
The rate at which heat energy flows through the condenser to the atmosphere is 158,775 kJ/min. This result is based on the assumption that all the heat lost by the steam is transferred to the cooling water, which then carries the heat to the atmosphere. The difference between the two values (158,775 kJ/min and 148,000 kJ/min) might be attributed to system inefficiencies, energy losses, or assumptions within the question setup.