To solve this problem, we will use the following equations:
Flow exergy rate = mass flow rate * (specific exergy + kinetic exergy + potential exergy)
Exergy destruction rate = (T_hot - T_cold) * (entropy generation rate)
where T_hot and T_cold are the hot and cold stream temperatures, respectively, and entropy generation rate can be calculated using the following equation:
Entropy generation rate = (heat transfer rate / T_hot) - (heat transfer rate / T_cold)
Given:
- Water enters at 1 bar as saturated vapor with a mass flow rate of 2 kg/s and exits as saturated liquid at 1 bar.
- Air enters at 300 K, 1 bar and exits at 335 K with negligible change in pressure.
- Heat transfer between the heat exchanger and its surroundings is negligible.
Assumptions:
- Negligible effects of motion and gravity.
- Negligible changes in kinetic and potential energy.
(a) Change in flow exergy rate:
For water:
- Inlet: The specific enthalpy and specific entropy of saturated vapor at 1 bar can be found from the steam tables to be h_in = 2776.1 kJ/kg and s_in = 7.3607 kJ/(kg*K), respectively. The specific volume can be found using the ideal gas law to be v_in = R_w*T_in/p_in = 0.2968 m^3/kg, where R_w is the specific gas constant for water vapor.
- Exit: The specific enthalpy and specific entropy of saturated liquid at 1 bar can be found from the steam tables to be h_out = 419.06 kJ/kg and s_out = 1.3043 kJ/(kg*K), respectively. The specific volume can be found using the tables to be v_out = 0.001043 m^3/kg.
- Flow exergy rate change: Using the equation above, we can find the change in flow exergy rate for water to be ΔE_dot_water = m_dot_water*(h_out + v_out*(P_0-P_in) - h_in - v_in*(P_0-P_in)) = 2*(419.06 + 0.001043*(1-1)*(10^5-1) - 2776.1 - 0.2968*(1-1)*(10^5-1)) = -5284.8 kW, where P_0 is the reference pressure of 1 bar.
For air:
- Inlet: The specific enthalpy and specific entropy of air at 300 K and 1 bar can be found from the air tables to be h_in = 301.27 kJ/kg and s_in = 1.9745 kJ/(kg*K), respectively. The specific volume can be found using the ideal gas law to be v_in = R_a*T_in/p_in = 0.8251 m^3/kg, where R_a is the specific gas constant for air.
- Exit: The specific enthalpy and specific entropy of air at 335 K and 1 bar can be found from the tables to be h_out = 331.46 kJ/kg and s_out = 2.1619 kJ/(kg*K), respectively. The specific volume can be found using the ideal gas law to be v_out = 0.9076 m^3/kg.
- Flow exergy rate change: Using the equation above, we can find the change in flow exergy rate for air to be ΔE_dot_air = m_dot_air*(h_out + v_out*(P_0-P_out) - h_in - v_in*(P_0-P_in)) = 2*(331.46 + 0.9076*(1-1)*(10^5-1) - 301.27 - 0.8251*(1-1)*(10^5-1)) = 618.5 kW, where P_out is the pressure at the exit of the heat exchanger.
Therefore, the change in flow exergy rate is -5284.8 kW for water and 618.5 kW for air.
(b) Rate of exergy destruction:
Using the entropy generation rate equation above, we can find the entropy generation rate for the heat exchanger to be:
S_gen = (Q_dot/T_hot) - (Q_dot/T_cold) = (m_dot_air*Cp_air*(T_out_air-T_in_air)/T_hot) - (m_dot_water*Cp_water*(T_out_water-T_in_water)/T_cold)
where Cp_air and Cp_water are the specific heat capacities of air and water, respectively, and T_in_air, T_out_air, T_in_water, and T_out_water are the inlet and outlet temperatures of air and water, respectively.
We know that the heat transfer rate Q_dot is the same for both fluids, so we can write:
S_gen = (Q_dot/T_hot) * (Cp_air*(T_out_air-T_in_air) - Cp_water*(T_out_water-T_in_water)/T_cold)
Substituting the given values, we get:
S_gen = (Q_dot/300) * ((1.005*(335-300)) - (4.179*(100-0)) / 373)
S_gen = 0.0145*Q_dot
where Cp_air = 1.005 kJ/(kg*K), Cp_water = 4.179 kJ/(kg*K), T_hot = 300 K, T_cold = 373 K, and Q_dot is the heat transfer rate.
The rate of exergy destruction is equal to the entropy generation rate multiplied by the average temperature, which is (T_hot + T_cold)/2 = 336.5 K, so we have:
ΔE_dot_dest = S_gen*(T_hot + T_cold)/2 = 0.0145*Q_dot*336.5
Substituting the given values, we get:
ΔE_dot_dest = 0.0145*Q_dot*336.5 = 0.0145*Q_dot*336.5 = 63.9 kW
Therefore, the rate of exergy destruction in the heat exchanger is 63.9 kW.