2F-3 : | Determination of Pressure Inside a Tank Containing Ammonia | 5 pts |
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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: |
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a.) Ideal
Gas EOS b.) Virial EOS c.) van der Waal EOS d.) Soave-Redlich-Kwong EOS e.) Compressibility Factor EOS f.) Ammonia Tables. |
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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: |
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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 |