Domain: Thermodynamics / heat recovery

An industrial furnace uses a heat recovery unit to improve fuel efficiency by transferring thermal energy from the exhaust gases to the incoming combustion air. The exhaust stream passes through this unit at a steady rate of 10 kg/s and cools from 300 °C to 200 °C. Over this temperature interval, its thermal properties may be treated as uniform, with a constant specific heat of 1.09 kJ/kg·K.

At the same time, fresh air enters the unit at 9 kg/s and an initial temperature of 40 °C. The air may be assumed to behave ideally, with a constant specific heat of 1.005 kJ/kg·K.

The energy exchange process is assumed to proceed without irreversibilities and without any shaft work, while the surroundings remain at a uniform temperature of 300 K. Pressure effects may be neglected.

Using only the basic governing principles applicable to steady-flow thermal systems under reversible conditions, determine the temperature reached by the air at the outlet of the heat recovery unit. Express the result in kelvin.

Grading rubric:
criterion: Correctly converts all process temperatures (both inlet and outlet for gas and air) from Celsius to absolute Kelvin and applies these to  isobaric entropy change model for constant specific heat ($\Delta S = m c_p \ln(T_2/T_1)$) rather than an enthalpy or internal energy model.
weight: 10
criterion: Correctly evaluates the specific entropy decrease of the flue gas stream (Delta S_g) explicitly as a negative value (approx. -2.09 kJ/K*s) and determines that the "reversible" and "steady-flow" conditions imply zero total entropy generation (Delta S_gen = 0 or Delta S_total = 0).
weight: 10
criterion: Correctly formulates the entropy balance equation where the entropy gain of the air exactly offsets the entropy loss of the flue gas (Delta S_air = -Delta S_gas).
weight: 20
criterion: Correctly isolates the natural logarithm term for the air temperature ratio (ln(T_out/313)) as approximately 0.231.
weight: 20
criterion: Correctly eliminates the natural logarithm using the exponential function (e^0.231) to estimate the temperature ratio (approx. 1.26).
weight: 10
criterion: Correctly solves for the final air outlet temperature (T_a,out) as approximately 394.4 K.
weight: 20
criterion: Correctly expresses the final result strictly in Kelvin as requested, ensuring no conversion back to Celsius.
weight: 10

Answer:
394.4 K