Exercises chapter 001. Volume / specific volume
1.1 Plotting Vapour Pressure
So that you get used to the different properties, we are going to spend some time plotting
them and commenting on them.
(a) You are to begin by plotting the vapour pressure of water against temperature. You need
the first steam table for this.
(b) You already know that ‘water boils at 100 °C'. Does this graph agree with your experience?
2.0
р
MPa
1.5
Sol.: In class.
1.0
0.5
T
0.0
0
100
200 °C
university of
groningen
CHTT-09. Technische thermodynamica / 1 1.2 Volume versus Pressure
Your second assignment: find all the data that you can on the volume of both steam and water
at a temperature of 200 °C. You will need data on 'saturated' steam and water, on
'superheated' steam and on compressed water. Plot these data in the logarithmic graph below
and answer the following questions:
(a) You will find that the vapour volume decreases with increasing pressure. The slope of the
line is -1; what does this mean? Remember that we are plotting the logarithm of volume
against the logarithm of pressure.
(b) What can you say about the effect of pressure on liquid volume?
(c) The line of the gas volumes has an abrupt step connection to the line of liquid volumes. At
which pressure does this occur?
Sol. In class.
10
m³ kg
1
0.1
T=200°C
0.01
р
MPa
0.001
0.01
0.1
1
10
university of
groningen
CHTT-09. Technische thermodynamica / 2 1.3 Volume versus Temperature
(1) Plot the volumes of liquid against temperature in the upper diagram. The pressure is 5
MPa. What do you see for temperatures up to 100 °C?
(2) Plot the volume of the gas against temperature in the lower diagram. The pressure is 0.1
MPa What does the slope in the graph for the gas indicate?
liquid at p=5MPa
0.001
0
m³ kg¹
T
°C
0.000
0
100
Sol.: In class.
10
vapour at p=0.1MPa
D
3
m² kg
T
K
1
100
1000
university of
groningen
CHTT-09. Technische thermodynamica / 3 1.4 Volume near the Critical Point
Your last plotting exercise. Do note that both scales in this logarithmic diagram have been
shifted by a factor of ten compared with that in exercise 1.2.
You are to plot the following:
the critical point
volume of the saturated vapour and of the saturated liquid
volume of the (superheated) vapour at 300 °C and at 400 °C
volume of the (compressed) liquid at 300 °C.
1
U
3
m³ kg³¹
0.1
0.01
0.001
Sol.: In class.
р
MPa
0.1
1
10
100
university of
groningen
CHTT-09. Technische thermodynamica / 4 1.5 Piping at Home
A suitable velocity for a liquid in a pipe is 1 m s-¹. For gases higher velocities are applied, say
10 m s-¹. Using these values calculate the inner diameters of pipes required for the following
purposes:
(a) Filling the flushing tank of a toilet. This requires about 6 L in one minute.
(b) For the forced ventilation of the house. This amounts to about one house volume per
hour, say 360 m³ h-¹.
Sol.:
(a) 1.1 cm;
(b) 11.3 cm.
1.6 Large Flows
Flows in technical systems can be much larger than in the household. Two examples:
(a) Suppose you need to drain an area of one square kilometre. The heaviest rainfall expected
is 1 cm h-¹. Which diameter ‘sewage pipe' would you need if the velocity in the pipe is 1 m
S-¹?
(b) A gas-fired power station might burn 100 m³ s-¹ of natural gas. This requires about 1000
m³ s-¹ of ambient air for combustion. (We study this in more detail in Lesson 10). This air
leaves the installation at a temperature of 450 K and ambient pressure (0.1 MPa). The
velocity in the stack (chimney) is 15 m s¹. What is the required stack diameter. (You may
find the result surprising.)
Sol.:
(a) 1.9 m;
(b) 11.3 m
1.7 Gas Pipeline
The main pipes of the National Gas Company have a diameter of 1.2 m. They are operated
under a pressure of 8 MPa, with a gas velocity of 5 m s-¹. You may assume a pipe temperature
of 290 K. How many moles of gas (methane CH4) are transported per second?
The gas constant has a value of 8.314 J mol¹ K-¹.
Sol.:
1.876× 104 mol/s.
1.8 Derivation of the simplified ideal gas law
The simplified ideal gas law establishes that pV/T is constant for a closed system. You are
asked to derive this expression from the general equation of state:
ov
dV =
ᎧᎢ
av
dT +
ӘР
dP
P
Taking into account the following historical empirical laws:
1) Boyle-Marriotte: At constant temperature the product pV is constant;
2) Gay-Lussac: At constant pressure the ratio V/T is constant.
Sol.: Challenge
1.9 Percentage of error using the ideal gas law
What is the percentage of error involved in treating steam as an ideal gas at the following
conditions?
a) p 1 MPa, T = 400 °C;
=
b) p = 10 MPa, T = 400 °C.
Sol.:
a) 1.3 %.
b) 17.8 %.
university of
groningen
CHTT-09. Technische thermodynamica / 5
