### Thermodynamics

# PVT Data from Compressibility Factor Chart

Nitrogen gas at a pressure of 100 Bar and -70^{o}C is contained in a tank of 0.25 m^{3}. Heat is added until the temperature is 37^{o}C. Determine approximately, through the use of Z factor,

**(a) Specific volume of gas, in m**

(b) The final pressure, in Bar.

^{3}/kmol(b) The final pressure, in Bar.

**Solution:**

*Data for N _{2}:*

Pc = 33.9 Bar

Tc = 126.2 K

The value of universal gas constant, consistent with the system of units in this problem is,

R = 1.01325 x 22.4/273 = 0.0831 Bar.m^{3}/(kmol.K)

**(a) Molar Volume of Gas:**

*Initial conditions:*

P_{1} = 100 Bar

T_{1} = (273 - 70) K

V_{1} = 0.3 m^{3}

P_{r1} = P_{1}/Pc = 100/33.9 = 2.9499

T_{r1} = T_{1}/Tc = 203/126.3 = 1.6082

Interpolating the value of Z for the T_{r1} and p_{r1} values, Z = 0.846

Molar volume of gas (v_{1}) is calculated as follows:
v_{1} = Z x R x T_{1}/P_{1} = 0.8460 x 0.0831 x 203 /100 = **0.1428 m ^{3}/kmol**

**(a) Molar Volume of Gas:**

*Final conditions:*

T2 = (273 + 37) K

v_{2} = v_{1}

P_{2}v_{2}/(RT_{2}) = Z_{2}.

In the above equation, both pressure and Z are unknowns.

To calculate Z at the final conditions, the following iterative procedure is used:

_{assumed}). For the first iteration the assumption shall be the ideal gas value (P

_{ideal}) obtained from the relation Pv = RT.

(ii) Using the assumed pressure of step (i), calculate Tr, Pr and the corresponding value of Z from tables/charts.

(iii) Evaluate P from P_{calculated} = Z x P_{ideal}.

(iv) If the calculated pressure of step (iii), is equal to that of that assumed in step (i), the problem is essentially solved, (i.e., the assumed pressure is the answer to the problem), else repeat steps (i) to (iii) till P_{assumed} is nearly equal to P_{calculated}.

The above procedure is applied to this problem and the calculations are as below:

*Iteration 1*:

P_{assumed} = P_{ideal} = R x T_{2}/v_{2} = 180.5 Bar

Pr = 180.5/33.9 = 5.3246; Tr = 310/126.2 = 2.4564.

For these Pr and Tr values, Z = 1.0432.

Therefore, P_{calculated} = 1.0432 x 180.5 = 188.3 Bar.

Since P_{calculated} <> P_{assumed}, let us assume another pressure value say, 185 Bar.

*Iteration 2*:

P_{assumed} = 185 Bar

Z = 1.0464

P_{calculated} = 188.88 Bar

*Iteration 3*:

P_{assumed} = 190Bar

Z = 1.0499

P_{calculated} = 189.51 Bar

*Iteration 4*:

P_{assumed} = 189.5Bar

Z = 1.0495

P_{calculated} = 189.4 Bar

In the fourth iteration, 'assumed' and 'calculated' pressures are almost equal. So the pressure at the final conditions when T = 37^{o}c = **189.5 Bar**