Example Problem with Complete Solution

2F-3 : Determination of Pressure Inside a Tank Containing Ammonia 5 pts
Ammonia is contained in a tank with a volume of 145 L.  Determine the pressure when 25 kg of ammonia are in the tank and the temperature of the ammonia is 150oC.
                   
Data : Tc
405.4
 K     ω
0.250
    
  Pc
1.13E+07
 Pa     MW
17.01
 g NH3 / mol NH3
Read : Not much to say here.
Given : m
25
 kg V
145
 L
T
150
 oC
0.145
 m3
423.15
 K R
8.314
 J/mol-K
Assumptions : - The ammonia can accurately be treated as an ideal gas is this system.
Ideal Gas EOS : Equation. Eqn 1 Solve for pressure : Equation. Eqn 2
 
We must determine the molar volume before we can use Eqn 2 to answer the question.
Use the definition of molar volume: Equation. Eqn 3 Where : Equation. Eqn 4
n
1469.72
 mol NH3 V
9.87E-05
 m3/mol
0.0987
 L/mol
Now, plug values back into Eqn 2.
Be careful with the units. P
3.57E+07
 Pa
P
35.7
 MPa
However, since the molar volume is 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.
Truncated Virial EOS : Equation. Eqn 5
We can estimate B using :
Equation.
Eqn 6
Equation. Equation.
Eqn 7 Eqn 8
Where : Equation. Eqn 9
We can solve Eqn 5 for P : Equation. Eqn 10
Plugging numbers into Eqns 9, 7, 8, 6, 5 and 10 (in that order) yields :
TR 1.044 B
-9.29E-05
 m3/mol
B0 -0.3110 Z
5.89E-02
B1 -0.0047 P
2.1
 MPa
van der Waal EOS : Equation. 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.
Equation. Eqn 12 Equation. Eqn 13
a
0.4229
 Pa-mol2/m6 b
3.72E-05
 m3/mol
Now, we can plug the constants a and b into Eqn 11 to determine the pressure.
P
13.8
 MPa
Redlich-Kwong EOS : Equation. Eqn 14
We can determine the values of a, b and α, which are constants that depend only on the chemical species in the system, from the following equations.
Equation.
Eqn 15
   Equation. Eqn 16
Now, plug values into Eqns 14 -16 :
a
8.62780
 Pa-m6-K1/2/mol2
b
2.577E-05
 m3/mol P
14.1
 MPa
Compressibility EOS : Given TR and the ideal reduced molar volume, use the compressibility charts to evaluate either PR or the compressibility, Z
Equation.
Eqn 17 Equation. Eqn 9
TR 1.0438
Defiition of the ideal reduced molar volume : Equation. Eqn 18
VRideal
0.3317
Read the Generalized Compressibility Chart for PR = 0 to 1 :
PR
1.25
 
Z
0.40
We can use the definition of PR to calculate P :
Equation. Eqn 19 Equation.
Eqn 20
 P
14.2
 MPa
Or, we can use Z and its definition to determine P : Equation. Eqn 21
P
14.3
 MPa
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 : Equation. Eqn 22
Using the MW of ammonia from part a yields : v
5.80E-06
 m3/g
v
5.80E-03
 m3/kg
The Superheated Ammonia Table gives us :
At P =
14
 MPa and At P =
16
 MPa
v =
0.0058196
 m3/kg v =
0.0035504
 m3/kg
We can determine the pressure in our tank by interpolation : P
14.0
 MPa
Verify: The ideal gas assumption made in the beginning of this problem solution was not valid !
Answers : P
14.0
 MPa