Compressibility of the fluid with density $\begin{array}{l}\rho\end{array}$ and molar volume $\begin{array}{l}V_m\end{array}$ as a function of temperature $\begin{array}{l}T\end{array}$ and pressure $\begin{array}{l}p\end{array}$ :

(1)	$\begin{array}{l}\displaystyle c(T,p) = \frac{1}{\rho} \left( \frac{\partial \rho}{\partial p} \right)_T = - \frac{1}{V_m} \left( \frac{\partial V_m}{\partial p} \right)_T\end{array}$

There is no universal analytical model for Fluid Compressibility but there is a good number of approximations which can be effectively used in engineering practice.

Approximations

Incompressible fluid

Compressible fluid

Full-Range Proxy Model

Slightly compressible fluid

Strongly Compressible Fluid

Real Gas

Ideal Gas

(2)	$\begin{array}{l}\displaystyle c(T, p) = 0\end{array}$

(3)	$\begin{array}{l}\displaystyle c(T, p) = c_0 = \rm const\end{array}$

...

(4)	$\begin{array}{l}\displaystyle c(T, p) = \frac{1}{p}\end{array}$

(5)	$\begin{array}{l}\displaystyle c(T, p) = \frac{c_0(T,p)}{1+c_0(T,p) \cdot p}\end{array}$

(6)	$\begin{array}{l}\displaystyle \rho(T, p) = \rho_0(T)\end{array}$

(7)	$\begin{array}{l}\displaystyle \rho(T, p) = \rho_0 \cdot \exp \left[ c_0(T) \cdot (p-p_0) \right]\end{array}$

...

(8)	$\begin{array}{l}\displaystyle \rho(T, p) = \frac{\rho_0(T)}{p_0} \cdot p\end{array}$

(9)	$\begin{array}{l}\displaystyle \rho(T, p) = \rho_0(T) \cdot \frac{1+c_0(T,p) \, p}{1+c_0(T,p) \, p_0}\end{array}$

(10)	$\begin{array}{l}\displaystyle Z(T, p) = \frac{p}{p_0}\end{array}$

(11)	$\begin{array}{l}\displaystyle Z(T, p) =\frac{p}{p_0}\cdot \exp \left[ - c_0(T) \cdot (p-p_0) \right]\end{array}$

...

(12)	$\begin{array}{l}\displaystyle Z(T, p) = 1\end{array}$

(13)	$\begin{array}{l}\displaystyle Z(T, p) = \frac{p}{p_0} \cdot \frac{1+c_0(T, p) \, p_0}{1 + c_0(T,p) \, p}\end{array}$

where

$\begin{array}{l}c\end{array}$	fluid compressibility
$\begin{array}{l}\rho\end{array}$	fluid density
$\begin{array}{l}Z\end{array}$	Z-factor

Mathematical models of Fluid Compressibility are reviewed in Fluid Compressibility @model.

The multi-phase fluid compressibility is a linear sum of compressibilities of its phases (see multi-phase fluid compressibility @ model).

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Fluid Compressibility

Approximations