01.10.
2024 5CHA0: Classical and Modern Physics
PROBLEMS Week 5
Homework
(submission deadline 08.10.2024 12:30)
H-5.1 Otto cycle. A gasoline engine has a volumetric compression ratio r of 8.0 and before
compression contains an ideal gas with cV = 0.717 kJ/(kg·K) and cP = 1.004 kJ/(kg·K) at
280 K and 85 kPa. The combustion generates a peak pressure of 6500 kPa. Find:
(a) the peak temperature,
(b) the energy (per unit mass) added by the combustion process,
(c) the exhaust temperature.
H-5.2 Portable air compressor. A heat-powered
q
portable air compressor consists of three components: an
adiabatic compressor, a constant-pressure heater (heat
supplied from an outside source) and an adiabatic turbine.
Ambient air enters the compressor at P1=100 kPa and
T1=300 K and is compressed to P2=600 kPa. Consider air
as an ideal gas, P v (cP cV )T with cP cV 1.40 .
The specific heat capacities are considered to be constant
for the temperature range of interest. For an adiabatic
process it was derived that Tv 1 K a and Pv K b , with Ka and Kb constants.
(a) The specific work for a steady state process is given by w vdP . Show, using this
expression, that for an adiabatic process from initial state i to final state j the specific
work equals w
1
Pi vi Pj v j cP (Ti T j ) .
1 1/
(b) Show for an adiabatic process that P 11 / T K c with Kc a constant.
We can use the results of a) and b) to analyse the portable compressor. All power from the
turbine goes into the compressor, and the turbine exhaust then supplies the compressed air.
(c) What is the temperature T2 at the exit of the compressor?
(d) If the desired exit pressure P4 is 200 kPa, give the temperature T3 at the exit of the heater.
H-5.3 Entropy. A 4.5 kg block of ice at 0.0 ºC falls into a large lake and melts. The
temperature of the water in the lake is 3.5 ºC. The lake is large enough to assume that its
temperature change can be neglected. The specific heat capacity of water is 4.18 kJ/(kg·K),
the specific heat of fusion is 330 kJ/kg.
(a) How much heat is extracted from the lake to melt the ice block and to turn it into water of
3.5 ºC?
(b) Calculate the change in entropy of the water in the lake. Does it increase or decrease?
(c) Calculate the change in entropy of the water from the original block of ice. Does the total
entropy (lake + melting ice block) increase or decrease?
Problems
1. The engine of a Ferrari F355 F1 sports car takes in air at 20.0°C and 1.00 atm. and
compresses it adiabatically to 0.090 times the original volume. The air may be treated as
an ideal gas with γ = 1.40.
(a) Draw a PV-diagram for this process.
(b) Find the final temperature and pressure.
2. I) The Otto cycle has four phases: adiabatic compression (1→2),
isochoric combustion (2→3), adiabatic power stroke (3→4), and
isochoric exhaust (4→1). At stage 1 the temperature is 15 ºC and
the pressure is 100 kPa. Assume that the fuel behaves as ideal
gas with cV = 0.717 kJ/(kg·K) and cP = 1.004 kJ/(kg·K).
(a) If the compression ratio r (= v1/v2) is 10.0 and added heat
during combustion qH is 1800 kJ/kg, calculate temperature
and pressure at stages 2, 3 and 4.
(b) Express the efficiency η in terms of temperatures T1 to T4 and
calculate its value for the temperatures calculated in (a).
1
(c) Show that the efficiency can be written as 1 1 ;
r
γ = cP/cV. Hint, show from the equal compression ratios in both adiabatic processes
that T3/T4 is equal to T2/T1, and apply this result to the expression found in (b).
II) For the Diesel cycle the combustion stroke (2→3) must be replaced by an isobaric
process.
(d) Show that the volume ratio during the power stroke (3→4) is given by
v4 v4 v2 T
r 2 .
v 3 v 2 v3 T3
(e) Stage 1 has 100 kPa and 15 °C. For ratio r = 20.0 and qH = 1800 kJ/kg find
temperature and pressure at stages 2, 3 and 4.
(f) Express the efficiency in terms of temperatures and calculate its value.
3. Assume a supply of 5 kg/s steam at 0.5 MPa is
needed for an industrial process. The setup in the
figure is used to extract the supply from the
high-pressure turbine, which at the same time
produces work (cogeneration). Energy from flow
velocity and height differences are neglected.
(a) Find the enthalpy values corresponding to
the three states 1-3 in the figure using:
https://app.knovel.com/sc/. States 1 and 2 are superheated steam; select “Compressed
Water and Superheated Steam”. State 3 is mixed vapour (fraction 0.90) and liquid
(remaining fraction of 0.10); select “Saturation and Quality Properties (by Pressure)”.
Mind the units.
(b) Find the power the turbine generates in this process. Assume the turbine is adiabatic.
(Hint: to find the exit enthalpy, take the weighted sum of state 2 and 3 according to
the mass flow rates.)
4. A small dam has a 0.5 m diameter pipe (at 1, see
figure) carrying liquid water at P1 = 150 kPa with a
flow rate of 2000 kg/s. The pipe runs to the bottom of
the dam 15 m lower into a turbine with pipe diameter
0.35 m (at 2). Assume no friction or heat transfer in y1 = 15 m
the complete system (internal energy water does not y2 = y3 = 0 m
change). Density of water is ρ = 1.00×103 kg/m3.
(a) Show that the 1st main law of thermodynamics for a continuous flow
q h1 12 v12 gy1 h2 12 v22 gy2 w for the flow from 1 to 2 reduces to the
Bernoulli equation.
(b) Calculate the flow velocities v1 and v2 at points 1 and 2 and the pressure P2 at the
turbine inlet.
(c) If the turbine exhausts (at 3) to P3 = 100 kPa, what is the power of the turbine (assume
that at 3 the flow velocity is almost 0)?
5. A jet engine has temperature after combustion
(isobaric) of 1500 K (state 3, see figure). The
compressor inlet is at 80 kPa, 260 K (state 1). The
combustors add 750 kJ/kg of specific heat.
Compressor and turbine are adiabatic and have no
friction; further neglect kinetic energy of the flow
except out of the nozzle. Pressure at the exit of
the nozzle is 80 kPa. Use the specific heat
capacities (constants) for air: cV = 0.717 kJ/(kg·K)
and cP = 1.004 kJ/(kg·K).
(a) Find temperature at state 2, the exit of the compressor.
(b) Find the pressure at the combustors (states 2 and 3).
(c) Why are the temperature differences across compressor and turbine equal? Find T4.
(d) What is the nozzle exit velocity?
6. Energy can be removed from water as heat at even below the normal freezing point
(0.0 ºC at atmospheric pressure) without causing the water to freeze; the water is then
supercooled. Suppose a 1.0 g water drop is supercooled until its temperature is that of the
surrounding air, which is at -5.0 ºC. The drop then suddenly and irreversibly freezes,
transferring energy to the air as heat.
(a) For the calculation of the entropy change of the drop, why is it allowed to replace this
irreversible freezing process by a reversible path?
(b) Replace the freezing at -5.0 ºC by a three-step reversible process: (i) increase the
liquid drop temperature to the normal freezing point; (ii) freeze it at 0.0 ºC to ice; (iii)
cool the ice back to -5.0 ºC. Sketch the three steps of this process in a T-S diagram.
(c) Calculate the entropy change of the drop (properties of water are listed below).
Properties of water
ρliq = 1.00×103 kg/m3; cliq = 4.18 kJ/(kg·K); cice = 2.22 kJ/(kg·K), lf = 330 kJ/kg
Self-study problems “Understanding Physics”
1st and 2nd edition:
Chapter 11 Worked examples: 11.4
Problem: 11.11, 11.12
3rd edition:
Chapter 11 Worked examples: 11.4
Problem: 11.14.x
Answers problems
1a) --- 1b) 495 ˚C, 29.1 atm.
2a) Stage 1 Stage 2 Stage 3 Stage 4
T (K) 288 724 3235 1287
P (MPa) 0.100 2.51 11.23 0.447
T T
2b) 1 4 1 60.2% 2c) --- 2d) ---
T3 T2
2e) Stage 1 Stage 2 Stage 3 Stage 4
T (K) 288 956 2749 1265
P (MPa) 0.100 6.63 6.63 0.439
1 T4 T1
2f) 1 61.1%
T3 T2
3a) 3375.06 kJ/kg, 2755.67 kJ/kg, 2373.20 kJ/kg 3b) 18.12 MW
4a) --- 4b) 10.2 m/s, 20.8 m/s, 132.8 kPa 4c) 498 kW
5a) 753 K 5b) 3301 kPa 5c) 1007 K 5d) 991 m/s
T
6a) --- 6b) (ii) S 6c) -1.17 J/K
0°C
(iii) (i)
-5°C
Figures are taken from “Fundamentals of Thermodynamics, 6th and 8th edition © John Wiley
& Sons, Inc.
