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 Urieli is licensed under a
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