Politecnico di Milano
Department of Energy - School of Industrial Engineering
Course Energy Systems
prof. M. GATTI – Academic Year 2023/2024
Precept 1
Mass and energy balances of energy systems
1. Energy balances for open systems
A machine absorbs, from an external source, a thermal power of 60 MW and produces mechanical power
meant to move a propeller.
The machine uses dry air as working fluid, which can be treated as an ideal gas (Rair = 287 J/(kg K)). The input
and output sections are both 0.5 m2 and on both of them the pressure is 0.8 bar. At the input section, the
velocity and the temperature are, respectively, w1 = 200 m/s and T1 = 20°C. At the output section the air
temperature is 300°C. The enthalpy variation is h2 – h1 = 300 kJ/kg.
It is required:
a) To calculate the velocity of the air of the output section.
b) To determine the power available at the shaft of the propeller.
2. Energy balance of a heat exchanger
In a nitrogen/water heat exchanger a stream of 3 kg/s of N2 is cooled from 500°C to 300°C and a stream of 3
kg/s of water enters at 15°C. Assume that the walls separating the heat exchanger from the external
environment are adiabatic. Moreover, assume the following data: MMN₂ = 28 kg/kmol; cL,H₂O = 4.186 kJ/(kg
K). Calculate:
a) The thermal power exchanged.
b) The water outlet temperature.
3. Energy balance of an open system
A thermodynamic system absorbs mechanical power from the external environment, to which,
simultaneously, it releases a thermal power equal to half of the absorbed mechanical power. The system is
fed with air (R = 287 J/(kg K); cP = 1004 J/(kg K)) at T1 = 15°C, P1 = 1 bar, through a duct with a diameter D1 =
0.1 m. In the inlet duct the velocity is w1 = 350 m/s. The outlet duct is placed at 45 m above the inlet duct
and is characterized by a diameter D2 = 0.15 m, moreover, the velocity and the pressure are, respectively, w2
= 100 m/s and P2 = 3 bar. Calculate:
a) The mechanical power.
b) The thermal power exchanged by the system.
4. Energy balance of a power plant
Two air streams exchange energy as in the diagram in Fig. 1, where the data of the problem can be found.
Assume that: the air behaves as a perfect gas with a specific heat cP = 1.005 kJ/(kg K); all the three
components (the two turbines and the heat exchanger) are adiabatic; the variations of kinetic and potential
energy of the air streams are negligible. It is required:
a) To calculate T3.
b) To determine the mechanical power produced by the Turbine 2.
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Due flussi d’aria differenti scambiano energia secondo lo schema riportato in Fig. 1, sul quale sono
inoltre riportati i dati del problema.
Assumendo che: l’aria si comporti come un gas perfetto avente calore specifico cp=1.005 kJ/(kgK);
tutti e tre i componenti (le due turbine e lo scambiatore di calore) siano adiabatici; le variazioni di
energia cinetica e potenziale dei flussi d’aria siano trascurabili.
Department of Energy - School of Industrial Engineering
È richiesto di:
Course Energy Systems
a) Calcolare T3 in °C.
prof.
M.
GATTI
Academic
b) Determinare la potenza meccanica prodotta dalla–turbina
2. Year 2023/2024
Politecnico di Milano
Turbine
1
Turbine
2
Air
in
heat exchanger
Air
in
Figura 1: Schema ed interazioni dei sistemi termodinamici oggetto dell’es. 2.3.
Figure 1: Diagram and interactions of the thermodynamic systems of exercise 4.
Pag. 1 di 2
5. Compression of an ideal gas
A compressor works under stationary conditions drawing ambient air temperature at 17°C and 1 bar. The air
is compressed up to 5 bar (outlet pressure). Kinetic and potential energy variations can be considered
negligible. Assuming that the machine works reversibly, it is required:
a) To calculate the work and the heat per unit of mass exchange in the following processes (each
corresponding to a different system):
i. Isothermal compression.
ii. Polytropic compression characterized by an exponent n = 1.3.
iii. Adiabatic compression.
b) To represent the processes in the p-v and T-s diagrams.
Compare and discuss the values of the work, heat and final temperatures determined for the three
considered cases.
6. Cooling with air dehumidification
A conditioner is fed with a stream of 10 m3/min of ambient air at P1 = 101.325 kPa, T1 = 30°C and with a
relative humidity equal to 80%. The air comes out of the air conditioner in saturated conditions at 14°C. The
water vapor that condenses as a result of the cooling leaves the conditioner at 14°C. Determine:
a) The flow rate of water vapor separating from the air.
b) The thermal power.
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Politecnico di Milano
Department of Energy - School of Industrial Engineering
Course Energy Systems
prof. M. GATTI – Academic Year 2023/2024
7. Mixture of ideal gases
Air is a mixture of N2 and O2 and small quantities of other gases; it can be approximated as a mixture of 79%
N2 and 21% O2 in terms of moles. During a stationary flow transformation, the air is cooled from 220 K to 160
K at a constant pressure of 10 MPa (Fig. 2). Determine the amount of heat exchanged (per kmol of air) during
the transformation considering the approximation of perfect gas.
Figure 2: Scheme of exercise 7.
8. Selecting and using the right thermodynamic model
Determine the specific volume of water vapor at 200 bar and 400 °C, using the following thermodynamic
tools or Equations of State (EoS):
a) Steam tables.
b) Ideal gas equation of state.
c) Compressibility chart.
d) Redlich–Kwong equation (EoS).
e) Van der Waals equation (EoS).
Represent the thermodynamic state on the p-T (pressure-temperature) and T-s (temperature-entropy)
diagrams.
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