We continue with our discussion on Pure Substances.

We find that for a pure substance in the superheated
region, at specific volumes much higher than that at the critical
point, the P-v-T relation can be conveniently expressed by the
**Ideal Gas Equation of
State** to a high degree of accuracy,
as follows:

P v = R T

where: R is constant for a particular substance and is called the

Gas Constant

Note that for the ideal gas equation both the pressure P and the temperature T must be expressed in absolute quantities.

Consider for example the *T-v* duagram
for water as shown below:

The shaded zone in the diagram indicates the region that can be represented by the Ideal Gas equation to an error of less than 1%. Note that at the critical point the error is 330%.

The gas constant R can be expressed as follows:

The three commonly used formats to express the Ideal Gas Equation of State are:

**Solved Problem 2.5**
- A piston-cylinder device contains 0.5 kg saturated liquid water
at a pressure of 200 kPa. Heat is added and the steam expands
at constant pressure until it reaches 300°C.

- a) Draw a diagram representing the process showing the initial and final states of the system.
- b) Sketch this process on a
*T-v*(temperature-specific volume) diagram with respect to the saturation lines, critical point, and relevant constant pressure lines, clearly indicating the initial and final states. - c) Using steam tables determine the initial temperature of the steam prior to heating.
- d) Using steam tables determine the final volume of the steam after heating
- e) Using the ideal gas equation of state determine the final volume of the steam after heating. Determine the percentage error of using this method compared to that of using the steam tables.

Note: The critical point data and the ideal
gas constant for steam can be found on the first page of the **steam tables**.

**Solution Approach:**

Even if questions a) and b) were not required, this should always be the first priority item in solving a thermodynamic problem.

c) Since state (1) is specified as saturated
liquid at 200 kPa, we use the
**saturated
pressure steam tables** to determine that T_{1}
= T_{sat@ 200kPa} = 120.2°C.

d) From the T-v diagram we determine that state
(2) is in the superheated region, thus we use the
**superheated
steam tables** to determine that v_{2} = v_{200kPa,300°C}
= 1.3162 m^{}3/kg. Thus V_{2} = m,v_{2} = (0,5kg).(1.3162
m^{}3/kg)
= 0.658 m^{}3 (658 liters)

Note that in doing a units check we find that
the following conversion appears so often that we feel it should
be added to our Units Conversion Survival Kit (recall
**Chapter
1**):

Finally we determine the percentage error of using the ideal gas equation at state (2)

**Problem 2.6** -
Consider a rigid container having a volume of 100 liters, filled
with steam at an initial state of 400 kPa and 300°C. The steam
is then cooled until it reaches a temperature of 90°C.

- a) Draw a diagram representing the process showing the initial and final states of the system.
- b) Using steam tables determine the mass of steam in the container. [0.153 kg]
- c) Using the ideal gas equation of state
determine the mass of steam in the container. [0.151 kg]

Determine the percentage error of using this method compared to that of using the steam tables. [1%] - d) Sketch this process on a
*T-v*(temperature-specific volume) diagram with respect to the saturation lines, critical point, and relevant constant pressure lines, clearly indicating the initial and final states. - e) Using steam tables determine the final pressure and quality of the fluid mixture after cooling. [70.2 kPa, X = 0.277]

Note: The critical point data and ideal gas
constant for steam can be found on the first page of the **steam tables**.

**Solved Problem 2.7**
- An automobile tire with a volume of 100 liters is inflated to
a gage pressure of 210 kPa. Determine a) the mass of air in the
tire if the temperature is 20°C, and b) the increase in gage
pressure if the temperature in the tire reaches 50°C. Assume
that atmospheric pressure is 100 kPa.

**Solution Approach:**

We always begin a thermodynamic problem with a sketch, indicating all the relevant information on the sketch, thus:

For part b) the temperature in the tire increases to 50°C (323K), however the volume and mass of air in the tire remains constant, thus:(Note for the SI challenged - initially the pressure was 30 psig, and then rose to 35 psig. Try to validate these values)

**Solved Problem 2.8**
- In aircraft design it is common practice to specify a standard
temperature distribution in the atmosphere near the surface of
the earth (up to an elevation z of 10000m) as T(z) = (T_{0}
+ a.z)°C, where T_{0} at the earth's surface is 15°C,
and a is the Temperature Lapse Rate (= -0.00651°C / m). Using
the Ideal Gas Equation of State, determine the pressure at an
elevation of 3000m if at z = 0, P = 101 kPa.

**Solution Approach:**

We first develop the solution in terms of the Hydrostatic Equation on an elemental height of the column of air, the Ideal Gas Equation of State, and the Temperature Lapse Rate equation, as follows:

**Solved Problem 2.9**
- You may wonder why we would be interested in knowing the value
of air pressure at 3000m altitude. In the following example we
continue with the above development in order to evaluate the payload
that can be lifted to an altitude of 3000m using a small hot air
balloon (Volume =1000 m^{}3) having an empty mass of 100kg. Assume that
the temperature of the air in the balloon is 100°C.

**Solution Approach:**

In this case we develop the solution in terms of a force balance between the bouyancy force (weight of the displaced air) and the gravity force including the weight of the hot air, the balloon empty mass, and the payload mass. The maximum altitude occurs when those two forces are equal, as follows:

Finally - with 154 kg payload
at least 2 persons can share and enjoy this wonderful experience.
Unfortunately they will not be able to enjoy a decent cup of English
tea. At a pressure of 69.9 kPa water will boil at (heavens forbid)
< 90°C! (Saturation temperature
T_{sat} from the
**Steam
Tables**). **Quick quiz:** determine the temperature of a cup of tea in Denver,
Colorado (elevation 5000 ft), or on the peaks of the Rocky Mountains
(elevation 14000 ft. *Hint*: use the Units Survival Kit presented
in **Chapter1** to first convert
from feet to meters)

________________________________________________________________

**Non-Ideal Gas Behavior**

We noticed in the above *T-v* diagram
for water that the gasses can deviate significantly from the ideal
gas equation of state in regions nearby the critical point and
there have been many equations of state recommended for use to
account for this non-ideal behaviour. However, this non-ideal
behaviour can be accounted for by a correction factor called the
**Compressibility Factor** Z defined as follows:

thus when the compressibility factor Z approaches 1 the gas behaves as an ideal gas. Note that under the same conditions of temperature and pressure, the compressibility factor can be expressed as:

Different fluids have different values of critical
point pressure and temperature data P_{CR} and T_{CR},
and these can be determined from the
**Table
of Critical Point Data of Various Substances.** Fortunately
the **Principle of Corresponding
States** states that we can normalize
the pressure and temperature values with the critical values as
follows:

All fluids normalized in this manner exhibit
similar non-ideal gas behaviour within a few percent, thus they
can all be plotted on a Generalised Compressibility Chart. A number
of these charts are available, however we prefer to use the **Lee-Kesler (logarithmic)
Compressibilty Chart**, The use of the compressibility chart
is shown in the following example.

**Solved Problem 2.10
-** Carbon Dioxide gas is stored in a
100 liter tank at 6 MPa and 30°C. Determine the mass of CO_{2}
in the tank based on (a) values obtained from the CO_{2}
tables of data, (b) the ideal gas equation of state, and (c) the
generalized compressibility chart.** **Compare (b) and (c)
to (a) and determine the percentage error in each case.

**Solution Approach:**

We first determine the Critical Point data
for CO_{2} from the **Table
of Critical Point Data of Various Substances**

After evaluating the Reduced Pressure and Reduced
Temperature we plot them on the **Generalized
Compressibility Chart** in order to determine the Compressibility
Factor, as shown below

The actual value of specific volume v_{a}
is obtained from the **CO _{2}
Superheat Tables**

The general rule is that if P << P_{CR}
or if T >> T_{CR} then you are probably dealing
with an ideal gas. If in doubt *always* check the Compressibility
Factor Z on the Compressibility Chart.

______________________________________________________________________________________

Engineering Thermodynamics by Israel Urieli is licensed under a
Creative Commons Attribution-Noncommercial-Share
Alike 3.0 United States License