Calculations

# Real Gas z Value

 Fill in the form below. Unit : kPa, K bar, K psi, R Gas : Other Air Nitrogen Methan Propane n-Butane Hydrogen Oxygen Carbon Dioxide (CO₂) Correlation : Hall and Yarborough’s correlation Kareem (2016) Dranchuk, Purvis, and Robinson Method Gas Pressure($$P$$) : bar Gas Temperature($$T$$) : K

Şekil :Standing and Katz’s compressibility factor chart

The z-factor is the ratio of the volume occupied by a given amount of a gas to the volume occupied by the same amount of an ideal gas:

$$z=\displaystyle\frac{V_{actual}}{V_{ideal}}$$

Substituting for $$V_{ideal}$$ in the ideal gas equation

$$PV_{actual}=znRT$$

For generalization, the z-factor is expressed as a function of pseudo-reduced temperature and pressure (Trube 1957; Dranchuk et al. 1971; Abou-kassem and Dranchuk 1975; Sutton 1985; Heidaryan et al. 2010).

Dranchuk et al. (1971) defined pseudo-reduced temperature and pressure as the ratio of temperature and pressure to the pseudo-critical temperature and pressure of natural gas, respectively:

$$T_{pr}=\displaystyle\frac{T}{T_{pc}}\,,\;\; P_{pr}=\displaystyle\frac{P}{P_{pc}}$$

The pseudo-critical properties of gas are the molar abundance (mole fraction weighted) mean of the critical properties of the constituents of the natural gas:

$$T_{pc}=\displaystyle\sum_{i=1}^n{y_iT_{c,i}}\,,\;\; P_{pc}=\displaystyle\sum_{i=1}^n{y_iP_{c,i}}$$

$$P$$  : Pressure (bar, kPa, psi)
$$P_{pc}$$ : Pseudo-critical pressure (bar, kPa, psi)
$$P_{pr}$$ : Pseudo-reduced pressure
$$P_{c,i}$$ : Gaz karışımındaki i.bileşenin kritik basıncı (bar, kPa, psi)

$$T$$  : Temperature (K, R)
$$T_{pc}$$ : Pseudo-critical temperature (K, R)
$$T_{pr}$$ : Pseudo-reduced temperature
$$T_{c,i}$$ : Gaz karışımındaki i.bileşenin kritik sıcaklığı (K, R)

$$y_{i}$$ : Gaz karışımındaki i.bileşenin molar yüzdesi

Şekil : Genel gaz P-T diyagramı

# $$z$$ Faktör Korelasyonları

The three most popular correlations for calculating the $$z$$-factor are implicit. The three correlations, described in the following subsections, are well known for their accuracies, almost unit correlation of regression coefficients and low maximum errors.

## Hall and Yarborough’s correlation (Trube 1957)

Implicit Z-factor correlation model for the natural gas in the wide range of pseudo-reduced temperature $$1.05 \leq T_{pr} \leq 3.0$$ and pseudo-reduced pressure $$0.2 \leq P_{pr} \leq 15$$

The $$z$$ value is obtained from the formula below.

$$z=\displaystyle\frac{A_1 P_{pr}}{y}$$

The value of $$y$$ is the root of the following equation and is obtained by iterating.

$$-A_1 P_{pr} +\displaystyle\frac{y+y^2+y^3-y^4}{(1-y)^3}-A_2y^2+\displaystyle A_3 \displaystyle y^{\displaystyle A_4}=0$$

Below are the values ​​$$A_1$$, $$A_2$$, $$A_3$$, $$A_4$$ in the equation.

$$A_1=0.06125t\displaystyle e^{-1.2 \left(1-t\right)^2}$$,
$$A_2=14.76t-9.76t^2+4.58t^3$$,
$$A_3=90.7t-242.2t^2+42.4t^3$$,
$$A_4=2.18+2.82t$$,
$$t= \displaystyle\frac{1}{T_{pr}}$$

## Dranchuk, Purvis and Robinson’s Correlation (Dranchuk et al. 1971)

Implicit Z-factor correlation @model for the natural gas in the wide range of pseudo-reduced temperature $$1.0 < T_{pr} \leq 3.0$$ and pseudo-reduced pressure $$0.2 \leq P_{pr} \leq 30$$ and also a specific range of $$0.7 < T_{pr} \leq 1.0$$ and $$P_{pr} < 1.0$$:

The $$z$$ value is obtained from the formula below.

$$z=\displaystyle\frac{0.27 P_{pr}}{yT_{pr}}$$

The value of $$y$$ is the root of the following equation and is obtained by iterating.

$$T_4y^2\left(1+A_{8}y^2\right)e^{-A_{8}y^2}+1+T_1y+T_2y^2+T_3y^5+\displaystyle\frac{T_5}{y}=0$$
$$T_1=A_1+\displaystyle\frac{A_2}{T_{pr}}+\displaystyle\frac{A_3}{T_{pr}^3}$$
$$T_2=A_4+\displaystyle\frac{A_5}{T_{pr}}$$
$$T_3=\displaystyle\frac{A_5A_6}{T_{pr}}$$
$$T_4=\displaystyle\frac{A_7}{T_{pr}^3}$$
$$T_5=\displaystyle\frac{0.27P_{pr}}{T_{pr}}$$

$$A_1=0.31506237$$, $$A_2=-1.04670990$$, $$A_3=-0.57832720$$,  $$A_4=0.53530771$$, $$A_5=-0.61232032$$, $$A_6=-0.10488813$$,  $$A_7=0.68157001$$, $$A_8=0.68446549$$

## Brill and Beggs’ compressibility factor (1973)

It is a relation that returns the value of $$z$$ directly without iteration.

$$z=A+\displaystyle\frac{1-A}{e^B}+C P_{pr}^D$$

Here,

$$A = 1.39(T_{pr} - 0.92)^{0.5} - 0.36T_{pr} - 0.10,$$
$$B = (0.62 - 0.23T_{pr} )p_{pr} + \left( {\displaystyle\frac{0.066}{{T_{pr} - 0.86}} - 0.037} \right)p_{pr}^{2} +\displaystyle \frac{{0.32p_{pr}^{2} }}{{10^{E} }},$$
$$C = 0.132 - 0.32\log (T_{pr} ),$$
$$D = 10^{F},$$
$$E = 9(T_{pr} - 1)\;,$$
$$F = 0.3106 - 0.49T_{pr} + 0.1824T_{pr}^{2}$$

## Dranchuk and Abou-Kassem’s correlation (Abou-kassem and Dranchuk 1975)

Implicit Z-factor correlation @model for the natural gas in the wide range of pseudo-reduced temperature $$1.0 < T_{pr} \leq 3.0$$ and pseudo-reduced pressure $$0.2 \leq P_{pr} \leq 30$$ and also a specific range of $$0.7 < T_{pr} \leq 1.0$$ and $$P_{pr} < 1.0$$:

The value of $$z$$ is obtained from the following formula.

$$z=\displaystyle\frac{0.27 P_{pr}}{yT_{pr}}$$

The value of $$y$$ is the root of the following equation and is obtained by iterating.

$$R_5y^2\left(1+A_{11}y^2\right)e^{-A_{11}y^2}+R_1y-\displaystyle\frac{R_2}{y}+R_3y^2-R_4y^5+1=0$$
$$R_1=A_1+\displaystyle\frac{A_2}{T_{pr}}+\displaystyle\frac{A_3}{T_{pr}^3}+\displaystyle\frac{A_4}{T_{pr}^4}+\displaystyle\frac{A_5}{T_{pr}^5}$$
$$R_2=\displaystyle\frac{0.27 P_{pr}}{T_{pr}}$$
$$R_3=A_6+\displaystyle\frac{A_7}{T_{pr}}+\displaystyle\frac{A_8}{T_{pr}^2}$$
$$R_4=A_9 \left( \displaystyle\frac{A_7}{T_{pr}}+\displaystyle\frac{A_8}{T_{pr}^2}\right)$$
$$R_5=\displaystyle\frac{A_{10}}{T_{pr}^3}$$

$$A_1=0.3265$$, $$A_2=-1.070$$, $$A_3=-0.5339$$,  $$A_4=0.01569$$, $$A_5=-0.05165$$, $$A_6=0.5475$$,  $$A_7=0.7361$$, $$A_8=0.1844$$, $$A_9=0.1056$$,  $$A_{10}=0.6134$$, $$A_{11}=0.7210$$

## Heidaryan, Moghdasi and Rahimi’s Correlation

It is a relation that returns the value of $$z$$ directly without iteration.

$$z = \ln \left( {\displaystyle\frac{{A_{1} + A_{3} \ln (P_{pr} ) + \displaystyle\frac{{A_{5} }}{{T_{pr} }} + A_{7} \left( {\ln (P_{pr} )} \right)^{2} + \displaystyle\frac{{A_{9} }}{{T_{pr}^{2} }} + \displaystyle\frac{{A_{11} }}{{T_{pr} }}\ln (P_{pr} )}}{{1 + A_{2} \ln (P_{pr} ) + \displaystyle\frac{{A_{4} }}{{T_{pr} }} + A_{6} \left( {\ln (P_{pr} )} \right)^{2} +\displaystyle \frac{{A_{8} }}{{T_{pr}^{2} }} + \displaystyle\frac{{A_{10} }}{{T_{pr} }}\ln (P_{pr} )}}} \right)$$

$$\begin{array}{ l l l } \rlap{\text{Tablw: Constants of the Heidarian correlation}}\\ \hline & P_{pr} \leq 3\; & P_{pr}>3\; \\ \hline \hline A_1 & \;\;\;2.827793 & \;\;\;3.252838 \\ A_2 & -4.688191x10^{-1} & -1.306424x10^{-1} \\ A_3 & -1.262288 & \;\;\;6.449194x10^{-1} \\ A_4 & -1.536524 & -1.518028 \\ A_5 & -4.535045 & -5.391019 \\ A_6 & \;\;\;6.895104 × 10^{-2} & -1.379588 × 10^{-2} \\ A_7 & \;\;\;1.903869 × 10^{-1} & \;\;\;6.600633 × 10^{-2} \\ A_8 & \;\;\;6.200089 × 10^{-1} & \;\;\;6.120783 × 10^{-1} \\ A_9 & \;\;\;1.838479 & \;\;\;2.317431 \\ A_{10} & \;\;\;4.052367 × 10^{-1} & \;\;\;1.632223 × 10^{-1} \\ A_{11} & \;\;\;1.073574 & \;\;\;5.660595 × 10^{-1} \\ \hline \end{array}$$

## Kareem $$z$$-faktor Correlation (2016)

$$z = \displaystyle\frac{{ DP_{pr} (1 + y + y^{2} - y^{3} )}}{{\left( {DP_{pr} + Ey^{2} - Fy^{G} } \right)(1 - y)^{3} }}$$

$$y = \displaystyle\frac{{ DP_{pr} }}{{\left( {\frac{{1 + A^{2} }}{C} - \displaystyle\frac{{A^{2} B}}{{C^{3} }}} \right)}} ,$$

Here,

$$t = \frac{1}{{T_{pr} }}$$,
$$A = a_{1} te^{{a_{2} (1 - t)^{2} }} P_{pr},$$  $$B = a_{3} t + a_{4} t^{2} + a_{5} t^{6} P_{pr}^{6},$$  $$C = a_{9} + a_{8} tP_{pr} + a_{7} t^{2} P_{pr}^{2} + a_{6} t^{3} P_{pr}^{3},$$  $$D = a_{10} te^{{a_{11} (1 - t)^{2} }},$$   $$E = a_{12} t + a_{13} t^{2} + a_{14} t^{3},$$  $$F = a_{15} t + a_{16} t^{2} + a_{17} t^{3},$$  $$G = a_{18} + a_{19} t$$

$$\begin{array}{ l l } \rlap{\text{Table:Constants of Kareem correlation}}\\ \hline \hline a_1 = 0.317842 & a_{11}= -1.966847 \\ a_2 = 0.382216 & a_{12} = 21.0581 \\ a_3 = -7.768354 & a_{13}= -27.0246 \\ a_4 = 14.290531 & a_{14}= 16.23 \\ a_5 = 0.000002 & a_{15} = 207.783 \\ a_6 = -0.004693 & a_{16} = -488.161 \\ a_7 = 0.096254 & a_{17}= 176.29 \\ a_8 = 0.166720 & a_{18}= 1.88453 \\ a_9 = 0.966910 & a_{19}= 3.05921 \\ a_{10}= 0.063069 & \\ \hline \end{array}$$

Bibliography : New explicit correlation for the compressibility factor of natural gas: linearized z-factor isotherms
Lateef A. Kareem, Tajudeen M. Iwalewa & Muhammad Al-Marhoun

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