Variation of Natural Gas Heat Capacity with Temperature, Pressure, and Relative Density

The change in enthalpy for a fluid where no phase change occurs between Points (1) and (2) can be expressed as:

(1)

The second term on the right hand side of this equation is generally not convenient to solve manually. However, it is trivial or zero for the following cases: (1) ideal gases, (2) constant pressure, dP = 0, and (3) for a liquid considered incompressible. For all three cases enthalpy is a mathematical function only of temperature. Cp is commonly expressed by equations of the form:

(2)

Where A, B, and C are constants that depend on system composition and T is the absolute temperature. In most instances it is sufficiently accurate to find a Cp at the average temperature TAvg, where:

(3)

CPAvg is then found at this average temperature and

(4)

This approximate solution to the first integral, although not exact, is satisfactory for most applications. Heat capacity values for pure substances are readily available from many handbooks and similar reference material. As noted in Chapter 7 of Volume 1, Gas Conditioning and Processing [1], values of heat capacity can be found from the slope of hvs. T plots at a given pressure. The CP for hydrocarbon liquid mixtures may be estimated from the equations presented in Volume 1 [1].
For a non-ideal, compressible fluid like natural gas, the second term on the right hand side of Eq.(1) can’t be ignored. Therefore, in process simulation software, an equation of state like Soave-Redlich-Kwong (SRK) [2] or Peng-Robinson (PR) [3] is used to calculate h. For many calculations involving the heat capacity of natural gas, Figure 8.3 in Volume 1 is appropriate. Heat capacity at system pressure and average temperature is read off the graph and multiplied by gas mass flow rate and T to obtain the heat load, .

(5)

In this Tip of The Month (TOTM), the variation of heat capacity of natural gases with temperature, pressure, and relative density (composition) will be demonstrated. Then an empirical correlation will be presented to account for these variations. This correlation will be used to estimate natural gas heat capacity for wide ranges of pressure, temperature, and relative density. Finally, the accuracy of the proposed correlation will be discussed.

Development of a Generalized CP Correlation:

As mentioned earlier, CP can be defined from the slope of h vs. T plots at constant pressure. Mathematically, this is expressed by:

(6)

The derivative on the right hand side of Eq (6) may be obtained from an equation of state (EOS) but it is too tedious for hand calculations. Therefore, the PR EOS option in ProMax [4] was used to generate CP values for various values of pressure, temperature, and relative density. The total number of CP values calculated was 715. Table 1 presents the composition of five different natural gas mixtures used in this study.
Table 1. Gas compositions used for generating CP values

Figures 1 through 5 present variations of CP with pressure, temperature and gas relative density. The red highlighted regions in Figures 3, 4, and 5 identify the two phase region of gas and liquid where the CP concept is not valid. It should be noted that the isobar of 20 MPa represents a single phase even at low temperatures. However, at low temperature, the fluid is dense phase.
In order to correlate all the curves shown in Figures 1-5 by a single equation, the following expression is proposed.

(7)

Where T is temperature, P is pressure and CP is heat capacity. A non-linear regression algorithm was used to determine the optimum values of parameters “a” through “f”. First, CP values of each gas in Table 1 were used to determine “a” through “f”. Then all of the generated CP values were used to determine a set of generalized parameters. These parameters were tuned and rounded to best represent all five gases covering a wide range of relative density from 0.60 to 0.80. For each case, the parameters and the summary of statistical error analysis are presented in Table 2. Note that the CP values of the two phase region were not used for the regression process. The general range of this correlation is from 20 to 200 °C (68 to 392 °F) and from 0.10 to 20 MPa (14.5 to 2900 Psia).

Discussion and Conclusions
A single and relatively simple correlation has been developed to estimate heat capacity of natural gases as a function of pressure, temperature, and relative density (composition). This correlation covers wide ranges of pressure (0.10 to 20 MPa, 14.5 to 2900 Psia), temperature (20 to 200 °C, 68 to 392 °F), and relative density (0.60 to 0.80). A generalized set of parameters in addition to an individual set of parameters have been determined and reported in Table 2. The error analysis reported in Table 2 indicates that the accuracy of this equation is quite good and can be used for natural gas heat duty calculations. For the generalized set of parameters, the average absolute percentage error (AAPD) and the maximum absolute percent deviations (MAPD) for the total of 715 points are 4.34 and 23.61, respectively. The applicable ranges of the proposed correlation are shown in Table 2.

Table 2. Parameters for the proposed correlation; Eq. (7) in SI and FPS system

AAPD= Average Absolute Percent Deviation and
MAPD= Maximum Absolute Percent Deviation
NPT= Number of Points and
SG = Relative Density (Specific Gravity)
Note: Below the above the temperature ranges for pressures 2, 5, 7, and 10 MPa (14.5 to 1450 Psia), the gas mixture may be in two phase (gas and liquid) region.

It should be noted that the concept of heat capacity is valid only for the single phase region.

Figures 3 through 5 indicate that for low temperatures, liquid forms and irregular behavior of CP is observed.
To learn more about similar cases and how to minimize operational problems, we suggest attending our G40(Process/Facility Fundamentals), G4 (Gas Conditioning and Processing) and G5 (Gas Conditioning and Processing – Special) courses.

By: Dr. Mahmood Moshfeghian

Reference:

1. Campbell, J. M., “Gas Conditioning and Processing, Vol. 1, the Basic Principals, 8th Ed., Campbell Petroleum Series, Norman, Oklahoma, 2001
2. G. Soave, Chem. Eng. Sci. 27 (1972) 1197-1203.
3. D.-Y. Peng, D.B. Robinson, Ind. Eng. Chem. Fundam. 15 (1976) 59-64.
4. ProMax®, Bryan Research & Engineering Inc, Version 2.0, Bryan, Texas, 2007