2F3 :  Determination of Pressure Inside a Tank Containing Ammonia  5 pts 

Ammonia
at 150^{o}C 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.) SoaveRedlichKwong EOS e.) Compressibility Factor EOS f.) Ammonia Tables. 

Data: T_{c} = 405.55 K, P_{c} = 11,280 kPa, MW = 17.03 g NH_{3}/mol NH_{3}, Pitzer accentric factor = 0.256.  
Read :  Not much to say here.  
Given :  m  7.4  kg  V  137  L  
T  150  ^{o}C  0.137  m^{3}  
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/molK  T_{c}  405.55  K  
MW  17.03  g NH_{3} / mol NH_{3}  P_{c}  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 NH_{3}  V  3.15E04  m^{3}/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 :  
T_{R}  1.043  B  9.34E05  m^{3}/mol  
B_{0}  0.3113  Z  7.04E01  
B_{1}  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  Pamol^{2}/m^{6}  b  3.74E05  m^{3}/mol  
Now, we can plug the constants a and b into Eqn 11 to determine the pressure.  
P  8.4  MPa  
Part d.)  
RedlichKwong 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  Pam^{6}K^{1/2}/mol^{2}  
b  2.590E05  m^{3}/mol  P  8.2  MPa  
Part e.)  
Compressibility EOS :  Given T_{R} and the ideal reduced molar volume, use the compressibility charts to evaluate either P_{R} or the compressibility, Z  

Eqn 17 

Eqn 9  
T_{R}  1.0434  
Defiition of the ideal reduced molar volume : 

Eqn 18  
V_{R}^{ideal}  1.055  
Read the Generalized Compressibility Chart for P_{R} = 0 to 1 :  P_{R}  0.70  
Z  0.73  
We can use the definition of P_{R} 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 > T_{c}, 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 ad) 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  1.85E05  m^{3}/g  
V  0.018514  m^{3}/kg  
The Superheated Ammonia Table gives us :  
At P =  7.5  MPa  and  At P =  10  MPa  
v =  0.020803  m^{3}/kg  v =  0.013381  m^{3}/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 