Example Problem with Complete Solution

2F-3 : Determination of Pressure Inside a Tank Containing Ammonia 5 pts
Ammonia at 150oC is contained in a tank with a volume of 137 L. The mass of the ammonia in the tank is 7.4 kg. Determine the pressure in the tank by each of the following methods:
a.) Ideal Gas EOS
b.) Virial EOS
c.) van der Waal EOS
d.) Soave-Redlich-Kwong EOS
e.) Compressibility Factor EOS
f.) Ammonia Tables.
Data: Tc = 405.55 K, Pc = 11,280 kPa, MW = 17.03 g NH3/mol NH3, Pitzer accentric factor = 0.256.
Read : Not much to say here.
Given : m 7.4 kg V 137 L
T 150 oC 0.137 m3
423.15 K
Find: P ??? kPa
Assumptions: None.
Equations / Data / Solve:
Begin by collecting all of the constants needed for all the Equations of State in this problem.
R 8.314 J/mol-K Tc 405.55 K
MW 17.03 g NH3 / mol NH3 Pc 1.128E+07 Pa
w 0.256
Part a.)
Ideal Gas EOS :
Eqn 1 Solve for pressure :
Eqn 2
We must determine the molar volume before we can use Eqn 2 to answer the question.
Use the definition of molar volume:
Eqn 3 Where :
Eqn 4
n 434.5 mol NH3 V 3.15E-04 m3/mol
0.3153 L/mol
Now, plug values back into Eqn 2.
Be careful with the units. P 1.12E+07 Pa
P 11.2 MPa
However, since the molar volume is FAR less than 20 L/mole, the Ideal Gas EOS is not applicable.
Choose any one of the following more sophisticated EOS's to solve the problem.
Part b.)
Truncated Virial EOS :
Eqn 5
We can estimate B using :
Eqn 6
Eqn 7
Eqn 8
Where :
Eqn 9
We can solve Eqn 5 for P :
Eqn 10
Plugging numbers into Eqns 9, 7, 8, 6, 5 and 10 (in that order) yields :
TR 1.043 B -9.34E-05 m3/mol
B0 -0.3113 Z 7.04E-01
B1 -0.0049 P 7.85 MPa
Part c.)
van der Waal EOS :
Eqn 11
We can determine the values of a and b, which are constants that depend only on the chemical species in the system, from the following equations.
Eqn 12
Eqn 13
a 0.4252 Pa-mol2/m6 b 3.74E-05 m3/mol
Now, we can plug the constants a and b into Eqn 11 to determine the pressure.
P 8.4 MPa
Part d.)
Redlich-Kwong EOS :
Eqn 14
We can determine the values of a, b and a, which are constants that depend only on the chemical species in the system, from the following equations.
Eqn 15
Eqn 16
Now, plug values into Eqns 14 -16 :
a 8.67636 Pa-m6-K1/2/mol2
b 2.590E-05 m3/mol P 8.2 MPa
Part e.)
Compressibility EOS : Given TR and the ideal reduced molar volume, use the compressibility charts to evaluate either PR or the compressibility, Z
Eqn 17
Eqn 9
TR 1.0434
Defiition of the ideal reduced molar volume :
Eqn 18
VRideal 1.055
Read the Generalized Compressibility Chart for PR = 0 to 1 : PR 0.70
Z 0.73
We can use the definition of PR to calculate P :
Eqn 19
Eqn 20
P 7.9 MPa
Or, we can use Z and its definition to determine P :
Eqn 21
P 8.1 MPa
Part f.) The Ammonia Tables provide the best available estimate of the pressure in the tank.
Because T > Tc, the properties of the ammonia in the tank must be obtained from the superheated vapor table, even though the it is actually a supercritical fluid in this system.
At this point we can make use of the fact that we have a pretty good idea of what the actual pressure is in the tank (from parts a-d) or we can scan the spuerheated vapor tables to determine which two pressures bracket our known value of the specific volume.
In either case, we begin by converting the molar volume into a specific volume :
Eqn 22
Using the MW of ammonia from part (a) yields : V 1.85E-05 m3/g
V 0.018514 m3/kg
The Superheated Ammonia Table gives us :
At P = 7.5 MPa and At P = 10 MPa
v = 0.020803 m3/kg v = 0.013381 m3/kg
We can determine the pressure in our tank by interpolation : P 8.3 MPa
Verify: No assumptions to verify.
Answers : a.) P 11.2 kPa d.) P 8.2 kPa
b.) P 7.9 kPa e.) P 8.1 kPa
c.) P 8.4 kPa f.) P 8.3 kPa