In this chapter we consider the property values and relationships of a pure substance (such as water) which can exist in three phases - solid, liquid and gas. We will not consider the solid phase in this course.

In order to introduce the rather complex phase change
interactions that occur in pure substances we consider an experiment
in which we have liquid water in a piston-cylinder device at 20°C
and 100kPa pressure.. Heat is added to the cylinder while the
pressure is maintained constant until the temperature reaches 300°C,
as shown in the following *T-v* diagram (temperature vs specific
volume):

From State (1) to State (2) the water maintains its
liquid phase and the specific volume increases very slightly until
the temperature reaches close to 100°C (State (2) - **Saturated
Liquid**). As more heat is added the
water progressively changes phase from liquid to water vapor (steam)
while maintaining the temperature at 100°C (**Saturation
Temperature** - T_{sat})
until there is no liquid remaining in the cylinder (State (4) -
**Saturated Vapor**).
If heating continues then the water vapor temperature increases (T >
T_{sat}) and is said
to be in the **Superheated**
(State (5)).

Notice that during this entire process the specific volume of the water increased by more than three orders of magnitude, which made it necessary to use a logarithmic scale for the specific volume axis.

We now consider repeating this experiment at various
pressures, as shown in the following *T-v* diagram:

Notice that as we increase the applied pressure, the
region between the saturated liquid and saturated vapor decreases
until we reach the **Critical
Point**, above which there is no clear
distinction between the liquid and vapor states.

It is common practice to join the loci of saturated
liquid and saturated vapor points as shown in the *T-v* diagram
below.

The saturation lines define the regions of interest
as shown in the diagram, being the **Compressed
Liquid **region, the **Quality
**region enclosed by the saturation
lines, and the **Superheat
**region (which also includes the
**Transcritical**
region) to the right of the saturated vapor line and
above the critical point. We will use **Property
Tables** associated with the regions in
order to evaluate the various properties. Notice that we have
provided property tables of steam, Refrigerant R134a, and Carbon
Dioxide, which we believe is destined to become the future
refrigerant of common usage.

The **Quality
Region** (also referred to as the **Saturated
Liquid-Vapor Mixture Region**) is
enclosed between the saturated liquid line and the saturated vapor
line, and at any point within this region the quality of the mixture
(also referred to as the dryness factor) is defined as the mass of
vapor divided by the total mass of the fluid, as shown in the
following diagram:

Notice that properties relating to the saturated liquid have the subscript f, and those relating to the saturated vapor have the subscript g. In order to evaluate the quality consider a volume V containing a mass m of a saturated liquid-vapor mixture.

Notice from the **steam
property tables** that we have also
included three new properties: internal energy u [kJ/kg], enthalpy h
[kJ/kg], and entropy s [kJ/kg.K] all of which will be defined as
needed in future sections. At this stage we note that the 3 equations
relating quality and specific volume can also be evaluated in terms
of these three additional properties.

The above discussion was done in terms of the *T-v*
diagram, however recall from Chapter 1 when we defined the State
Postulate that any two independent intensive properties can be used
to completely define all other intensive state properties. It is
often advantageous to use the *P-v* diagram with temperature as
the parameter as in the following diagram:

Notice that because of the extremely large range of
pressure and specific volume values of interest, this can only be
done on a log-log plot. This is extremely inconvenient, so both the
*T-v* and the *P-v* diagrams are normally not drawn to
scale, however are sketched only in order to help define the problem,
which is then solved in terms of the steam tables. This approach is
illustrated in the following solved problems.

**Solved Problem 2.1 - **Two
kilograms of water at 25°C are placed in a piston cylinder device
under 100 kPa pressure as shown in the diagram (State (1)). Heat is
added to the water at constant pressure until the piston reaches the
stops at a total volume of 0.4 m^{3}
(State (2)). More heat is then added at constant volume
until the temperature of the water reaches 300°C (State (3)).
Determine (a) the quality of the fluid and the mass of the vapor at
state (2), and (b) the pressure of the fluid at state (3).

**Step 1:****Always**
draw a complete diagram of the states and processes of
the problem and include all the relevant information on the diagram.
In this case there are three states and two processes (constant
pressure and constant volume).

__ Step 2:__ In the case
of a closed system with a phase change fluid,

Notice that the *T_v* diagram is based
exclusively on intensive properties, hence mass is not indicated on
the diagram. Thus we indicate on the diagram that in order to
determine the quality at state (2) we need to first evaluate the
specific volume v_{2}, which can then be compared to the
saturation values v_{f} and v_{g} at the pressure of
100 kPa.

Thus v_{2}= V / m = 0.4 [m^{3}] / 2 [kg] = 0.2 [m^{3}/ kg]

Concerning state (3), the problem statement did not
specify that it is in the superheat region. We needed to first
determine the saturated vapor specific volume v_{g}
at 300°C. This value is 0.0216 m^{3}
/ kg, which is much less than the specific volume v_{3}
of 0.2 m^{3}
/ kg, thus placing state (3) well into the superheated
region. Thus the two intensive properties which we use to determine
the pressure at state (3) are T_{3}
= 300°C, and v_{3} =
0.2 m^{3}
/ kg. On scanning the **superheat
tables** we find that the closest values
lie somewhere between 1.2 MPa and 1.4 MPa, thus we use linear
interpolation techniqes to determine the actual pressure P_{3}
as shown below:

**Solved Problem 2.2 - **Two
kilograms of water at 25°C are placed in a piston cylinder device
under 3.2 MPa pressure as shown in the diagram (State (1)). Heat is
added to the water at constant pressure until the temperature of the
fluid reaches 350°C (State (2)). Determine the final volume of the
fluid at state (2).

In this example since the pressure is known (3.2 MPa)
and remains constant throughout the process, we find it convenient to
draw a *P-v* diagram indicating the process (1) - (2) as
follows.

As in the previous example, on scanning the **superheat
tables** we find that we need to
interpolate between pressure P = 3.0 MPa and P = 3.5 MPa in order to
determine the specific volume at the required pressure of 3.2 MPa as
follows:

**Problem 2.3 - **A
piston-cylinder device contains a saturated mixture of steam and
water having a total mass of 0.5 kg at a pressure of 160 kPa and an
initial volume of 100 liters. Heat is then added and the fluid
expands at constant pressure until it reaches a saturated vapor
state.

a) Draw a diagram representing the process showing the initial and final states of the system.

b) Sketch this process on a

*P-v*diagram with respect to the saturation lines, critical point, and relevant constant temperature lines, clearly indicating the initial and final states.c) Determine the initial quality and temperature of the fluid mixture prior to heating. [quality x

_{1}= 0.182, T_{1}= 113.3°C]d) Determine the final volume of the steam after heating. [0.546 m

^{3}(546 liters)]

Note: 1000 liters - 1 m^{3}.

**Problem 2.4 - **A pressure
cooker allows much faster (and more tender) cooking by maintaining a
higher boiling temperature of the water inside. It is well sealed,
and steam can only escape through an opening on the lid, on which
sits a metal petcock. When the pressure overcomes the weight of the
petcock, the steam escapes, maintaining a constant high pressure
while the water boils.

Assuming that the opening under the petcock has an
area of 8 mm^{2},
determine

a) the mass of the petcock required in order to maintain an operating pressure of 99 kPa gage. [80.7gm]

b) the corresponding temperature of the boiling water. [120.2°C]

Note: Assume that the atmospheric pressure is 101 kPa. Draw a free body diagram of the petcock.

**On to Chapter 2b) of Pure
Substances**

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Engineering Thermodynamics by Israel
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