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 theGas 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* diagram 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