**1. Introduction**

In the last six decades, petroleum engineers realized the importance of using EOS for PVT modeling in addition to the following [1]:


### **1.1 Classification of equation of state**

There are different types of EOS which fall into three categories:

#### *1.1.1 First class of EOS*

These equations are basically cubic equation of state. The cubic equations of state such as the Van der Waals [2], Redlicha and Kwong [3], Soave-Redlich-Kwong [4], and Peng-Robinson [5] equations give reasonable results for the thermodynamic behavior of real fluids.

gases, the reason for this deviation is that the ideal gas law was derived under the assumption that the volume of molecules is very small and neither molecular attraction nor repulsion exists between them, and this is not the real case. In order to write an equation of state for a real gas, a correction factor has to be inserted into

where Z: the correction factor which is known as the compressibility factor. The equation has different names, such as the compressibility equation and/or the real gas equation [13]. A review of recent advances in the empirical cubic EOS field is presented next [11]. Van der Waals [2] is one of the earliest attempts to represent the behavior of real gases by an equation, where the two assumptions

1.The gas molecule volume is very small compared to the volume of the

2.There are no attractive or repulsive forces between the gas molecules or the

Van der Waals attempted to eliminate these assumptions in the development of

*First assumption elimination*: the gas molecules occupy a considerable fraction of the volume at higher pressures, and the volume of the molecules (b) is subtracted

*<sup>p</sup>* <sup>¼</sup> RT

*Second assumption elimination*: he added corrective term (a), denoted by (a/V2

The symbol "a" is considered a measure of the intermolecular attractive forces between the molecules. "b" is known as the co-volume and considered to reflect the volume of molecules [2]. The "a" and "b" values can be obtained from the critical properties of the fluid [14], where the repulsion pressure, prepulsion, is represented by the term RT/(Vm – b), and the attraction pressure, pattraction, is described by a/

**.** The Van der Waals equation of state despite its simplicity, while it is provide a correct description and qualitative of the PVT substances behavior in the liquid and gaseous phases. Yet, it is not accurate enough to be suitable for design purposes. The equation of state approach for calculating physical properties and phase equilibrium proved to be a powerful tool, and much energy was devoted to the development of new and accurate equations of state [11]. Other researchers began attempts to improve Van der Waals equation of state for over 100 years. Usually a change of the

that the molecular attraction term was inversely proportional to temperature [16]:

from the actual molar volume (V) to give the following expression**:**

in order to account for the attractive forces between molecules. Van der Waals introduced the following equation (Eq. (4)):

p þ

where a: attraction parameter; b: repulsion parameter.

a VM 2

P V ¼ Zn R T (2)

<sup>v</sup> � <sup>b</sup> (3)

ð Þ¼ V**<sup>M</sup>** � b RT (4)

) was proposed. Clausius in 1880 [15] proposed

),

the ideal gas equation [12]:

*DOI: http://dx.doi.org/10.5772/intechopen.89919*

*Equation of State*

were made for the ideal gas EOS**:**

walls of the container.

an empirical EOS for the real gases.

molecular attraction term (a/VmM2

container.

Vm 2

**147**

#### *1.1.2 Second class of EOS*

These EOS are non-cubic in form. They are providing accurate results for both vapor and liquid phases. The Benedict et al. [6] equation is a good example for this class equation.

#### *1.1.3 Third class of EOS*

These are nonanalytical EOS that are highly constrained for some specific fluids [7]. Even though they are constrained, they are capable of expressing real fluid thermodynamic properties precisely.

Among all these EOS, the first-class EOS is more useful because it provides an analytical solution than the more complex and complicated non-cubic second type and nonanalytical third type that require time-consuming iterative calculations. In general, the overall performance in fluid properties prediction is somewhat better using the Soave-Redlich-Kwong (SRK) equation than using the Redlich-Kwong (RK) and Van der Waals EOS [8].

### **2. Development history of the equation of state**

Several forms of EOS have been presented to the petroleum industry to estimate hydrocarbon reservoir fluid properties and sought to a better representation of the PVT relationship for fluids [9].

In 1662, Robert Boyle (Boyle's law) discovered that for a constant temperature, there is an inverse relationship between volume of gas and its pressure (P ∝ V�<sup>1</sup> ). In 1780, Jacques Charles (Charles's Law) showed that the volume of gas is proportional to the absolute temperature at a constant pressure (V ∝ T). In 1834, Clapeyron combined these two results into the ideal gas law, PV = RT [10], assuming that the molecules are very far and have no attractive or repulsive forces between them and elastic collisions between these molecules. This equation is known as the ideal gas law and/or the general gas law. It is expressed mathematically as [11].

$$\mathbf{P}\,\mathbf{V} = \mathbf{n}\,\mathbf{R}\,\mathbf{T} \tag{1}$$

where P: absolute pressure, psia; V: volume, ft3 ; T: absolute temperature, °R; R: the universal gas constant (10.73159 ft<sup>3</sup> psia °R�<sup>1</sup> lb-mole�<sup>1</sup> ; n: number of moles of gas, lb-mole.

For gases at low pressures, the ideal gas law is a convenient satisfactory tool. The application of the ideal gas law at higher pressures may lead to errors up to 500%, compared to 2–3% at atmospheric pressure. Real gases behave differently than ideal **1.1 Classification of equation of state**

*Inverse Heat Conduction and Heat Exchangers*

*1.1.1 First class of EOS*

namic behavior of real fluids.

*1.1.2 Second class of EOS*

*1.1.3 Third class of EOS*

thermodynamic properties precisely.

(RK) and Van der Waals EOS [8].

PVT relationship for fluids [9].

gas, lb-mole.

**146**

**2. Development history of the equation of state**

where P: absolute pressure, psia; V: volume, ft3

the universal gas constant (10.73159 ft<sup>3</sup> psia °R�<sup>1</sup> lb-mole�<sup>1</sup>

class equation.

There are different types of EOS which fall into three categories:

These equations are basically cubic equation of state. The cubic equations of state such as the Van der Waals [2], Redlicha and Kwong [3], Soave-Redlich-Kwong [4], and Peng-Robinson [5] equations give reasonable results for the thermody-

These EOS are non-cubic in form. They are providing accurate results for both vapor and liquid phases. The Benedict et al. [6] equation is a good example for this

These are nonanalytical EOS that are highly constrained for some specific fluids [7]. Even though they are constrained, they are capable of expressing real fluid

Among all these EOS, the first-class EOS is more useful because it provides an analytical solution than the more complex and complicated non-cubic second type and nonanalytical third type that require time-consuming iterative calculations. In general, the overall performance in fluid properties prediction is somewhat better using the Soave-Redlich-Kwong (SRK) equation than using the Redlich-Kwong

Several forms of EOS have been presented to the petroleum industry to estimate hydrocarbon reservoir fluid properties and sought to a better representation of the

In 1662, Robert Boyle (Boyle's law) discovered that for a constant temperature, there is an inverse relationship between volume of gas and its pressure (P ∝ V�<sup>1</sup>

For gases at low pressures, the ideal gas law is a convenient satisfactory tool. The application of the ideal gas law at higher pressures may lead to errors up to 500%, compared to 2–3% at atmospheric pressure. Real gases behave differently than ideal

P V ¼ n R T (1)

; T: absolute temperature, °R; R:

; n: number of moles of

1780, Jacques Charles (Charles's Law) showed that the volume of gas is proportional to the absolute temperature at a constant pressure (V ∝ T). In 1834, Clapeyron combined these two results into the ideal gas law, PV = RT [10], assuming that the molecules are very far and have no attractive or repulsive forces between them and elastic collisions between these molecules. This equation is known as the ideal gas

law and/or the general gas law. It is expressed mathematically as [11].

). In

gases, the reason for this deviation is that the ideal gas law was derived under the assumption that the volume of molecules is very small and neither molecular attraction nor repulsion exists between them, and this is not the real case. In order to write an equation of state for a real gas, a correction factor has to be inserted into the ideal gas equation [12]:

$$\mathbf{P}\,\mathbf{V} = \mathbf{Z}\mathbf{n}\,\mathbf{R}\,\mathbf{T} \tag{2}$$

where Z: the correction factor which is known as the compressibility factor.

The equation has different names, such as the compressibility equation and/or the real gas equation [13]. A review of recent advances in the empirical cubic EOS field is presented next [11]. Van der Waals [2] is one of the earliest attempts to represent the behavior of real gases by an equation, where the two assumptions were made for the ideal gas EOS**:**


Van der Waals attempted to eliminate these assumptions in the development of an empirical EOS for the real gases.

*First assumption elimination*: the gas molecules occupy a considerable fraction of the volume at higher pressures, and the volume of the molecules (b) is subtracted from the actual molar volume (V) to give the following expression**:**

$$p\_{\parallel} = \frac{\text{RT}}{\text{v} - \text{b}} \tag{3}$$

*Second assumption elimination*: he added corrective term (a), denoted by (a/V2 ), in order to account for the attractive forces between molecules.

Van der Waals introduced the following equation (Eq. (4)):

$$\left(\mathbf{p} + \frac{\mathbf{a}}{\mathbf{V\_M}^2}\right) \left(\mathbf{V\_M} - \mathbf{b}\right)^{\tag{4}} = \mathbf{R}\mathbf{T} \tag{4}$$

where a: attraction parameter; b: repulsion parameter.

The symbol "a" is considered a measure of the intermolecular attractive forces between the molecules. "b" is known as the co-volume and considered to reflect the volume of molecules [2]. The "a" and "b" values can be obtained from the critical properties of the fluid [14], where the repulsion pressure, prepulsion, is represented by the term RT/(Vm – b), and the attraction pressure, pattraction, is described by a/ Vm 2 **.** The Van der Waals equation of state despite its simplicity, while it is provide a correct description and qualitative of the PVT substances behavior in the liquid and gaseous phases. Yet, it is not accurate enough to be suitable for design purposes. The equation of state approach for calculating physical properties and phase equilibrium proved to be a powerful tool, and much energy was devoted to the development of new and accurate equations of state [11]. Other researchers began attempts to improve Van der Waals equation of state for over 100 years. Usually a change of the molecular attraction term (a/VmM2 ) was proposed. Clausius in 1880 [15] proposed that the molecular attraction term was inversely proportional to temperature [16]:

*Inverse Heat Conduction and Heat Exchangers*

$$\left[\mathbf{p} + \frac{\mathbf{a}}{\mathbf{T}(\mathbf{V\_M} + \mathbf{c})^2}\right](\mathbf{V\_M} - \mathbf{b}) = \mathbf{R}\mathbf{T} \tag{5}$$

The BWR equation could treat critical components and was able to work in the critical area. However, the BWR equation suffers from some disadvantages [25]. Perhaps, the most important model for the modification of the Van der Waals equation of state is the Redlich-Kwong (RK) (1949) which is demonstrated by an

temperature explicitly. They could improve the prediction of the physical and volumetric properties of the vapor phase. In RK EOS, the attraction pressure term was replaced with a generalized temperature-dependent term (Eq. (13)) [3]**:**

<sup>V</sup> � <sup>b</sup> � *<sup>α</sup>*

*For pure substances*, the equation parameters *a* and *b* are usually expressed as.

Tc 2*:*5

Replacing the molar volume (V) in Eq. (13) with (ZRT/P) and rearranging give.

*<sup>B</sup>* <sup>¼</sup> *bp*

*<sup>A</sup>* <sup>¼</sup> *ap* R2

Three real roots in the two-phase region are yielded. The largest root corresponds to the compressibility factor of the gas phase, Zv, while the smallest positive

*am* <sup>¼</sup> <sup>X</sup><sup>n</sup>

i¼1

*bm* <sup>¼</sup> <sup>X</sup><sup>n</sup> i¼1

am and bm for a hydrocarbon gas mixture with a composition of yi:

*<sup>α</sup><sup>m</sup>* <sup>¼</sup> <sup>X</sup><sup>n</sup>

*bm* <sup>¼</sup> <sup>X</sup><sup>n</sup> i¼1 *yi bi*

for the i component; bi: Redlich-Kwong b parameter for the i component; bm:

i¼1 *yi* ffiffiffiffi *ai* p " #<sup>2</sup>

where n: number of components in the mixture; ai: Redlich-Kwong a parameter

*For mixtures*, the equation parameters a and b are usually expressed as am and bm

*Xi* ffiffiffiffi *a*i p " #<sup>2</sup>

<sup>a</sup> <sup>¼</sup> Ω α R2

V Vð Þ <sup>þ</sup> <sup>b</sup> ffiffiffi

T

Z3 � <sup>Z</sup><sup>2</sup> <sup>þ</sup> <sup>A</sup> � <sup>B</sup> � <sup>B</sup><sup>2</sup> � � <sup>Z</sup> � AB <sup>¼</sup> <sup>0</sup> (16)

b ¼ ΩbRTc*=*Pc (14)

2

) and includes the system

p (13)

*=*Pc (15)

RT (17)

<sup>T</sup><sup>2</sup>*:*<sup>5</sup> (18)

*Xibi* ½ � (20)

� � (22)

(19)

(21)

adjustment of the Van der Waals's attraction term (a/Vm

*DOI: http://dx.doi.org/10.5772/intechopen.89919*

where Ω*a* = 0.42747 and Ωb = 0.08664.

root corresponded to that of the liquid, ZL [11].

for a hydrocarbon liquid mixture with a composition of xi:

where

*Equation of State*

**149**

*<sup>ρ</sup>* <sup>¼</sup> RT

The addition of a fourth constant (c) enabled better agreement with data. However, mathematical manipulations required in thermodynamic calculations were more difficult. So Berthelot in 1899 [17] removed the constant (c), resulting in the following equation:

$$\left(\mathbf{p} + \frac{\mathbf{a}}{\mathbf{T} \mathbf{V}\_{\mathbf{M}}^{2}}\right) \left(\mathbf{V}\_{\mathbf{M}} - \mathbf{b}\right) \tag{6}$$

Dieterici in 1899 [18] handled the temperature dependence of the molecular attraction term in a different manner [6]:

$$\left[\mathbf{P}\,\mathrm{EXP}\left(\frac{\mathbf{a}}{\mathbf{V\_{M}RT}}\right)\right](\mathbf{V\_{M}}-\mathbf{b})\,\,\,=\mathrm{RT}\tag{7}$$

Lorentz in 1881 [19] addressed the molecular volume term [20]:

$$\left(\mathbf{p} + \frac{\mathbf{a}}{\mathbf{V}\_{\mathcal{M}^2}}\right) \left(\mathbf{V}\_{\mathcal{M}} - \frac{\mathbf{b}\mathbf{V}\_{\mathcal{M}}}{\mathbf{V}\_{\mathcal{M}} + \mathbf{b}}\right) = \mathbf{RT}.\tag{8}$$

Wohl in 1927 [21] considered the effect of temperature on the molecular attraction term:

$$(P + \frac{a}{TV\_M(V\_M - b)} - \frac{c}{T^2 V\_M^{\;\;\;\;\beta}})(VM - b) = RT \tag{9}$$

The constants a, b, and c in the equations above have different values for different substances. Several investigations proposed virial-type of EOS. Kammerlingh-Onnes in 1901 [22] proposed the virial equation of state as follows [23]:

$$PV\_M = RT \left[ 1 + \frac{B}{V\_M} + \frac{C}{V\_M^{-2}} + \dots \right] \tag{10}$$

where B and C are not constants which are functions of temperature and are called the second and third virial coefficients. Beattie and Bridgeman in 1927 published a five-constant equation that gives a satisfactory representation of volumetric properties except in the critical region [24]:

$$P = \frac{RT}{V\_{M2}} \left( 1 - \frac{c}{V\_M T^3} \right) \left[ V\_M + B\_o \left( 1 - \frac{b}{V\_M} \right) \right] - \frac{A\_{o(1 - a/V\_M)}}{V\_M} \tag{11}$$

Benedict et al. [6] suggested a multiparameter equation of state known as the Benedict-Webb-Rubin (BWR) equation [6]:

$$P = \frac{RT}{V\_M} + \frac{B\_oRT - A\_o - C\_o/T^2}{V\_M^2} + \frac{bRT - a}{V\_M^3} + \frac{aa}{V\_M^6} + \frac{c}{T^2V\_M^{-3}} \left(1 + \frac{\gamma}{V\_M^{-2}}\right) \text{EXP} \left(\frac{-\gamma}{V\_M^{-2}}\right) \tag{12}$$

This equation may be considered a modification of the Beattie-Bridgeman equation of state where A0, B0, C0, a, b, c, α, and γ are eight adjustable parameters. The BWR equation could treat critical components and was able to work in the critical area. However, the BWR equation suffers from some disadvantages [25]. Perhaps, the most important model for the modification of the Van der Waals equation of state is the Redlich-Kwong (RK) (1949) which is demonstrated by an adjustment of the Van der Waals's attraction term (a/Vm 2 ) and includes the system temperature explicitly. They could improve the prediction of the physical and volumetric properties of the vapor phase. In RK EOS, the attraction pressure term was replaced with a generalized temperature-dependent term (Eq. (13)) [3]**:**

$$\rho = \frac{\text{RT}}{\text{V} - \text{b}} - \frac{a}{\text{V}(\text{V} + \text{b})\sqrt{\text{T}}} \tag{13}$$

*For pure substances*, the equation parameters *a* and *b* are usually expressed as.

$$\mathbf{b} = \mathbf{Q}\mathbf{b} \,\mathrm{R}\,\mathrm{T}\_{\mathrm{c}}/\mathrm{P}\_{\mathrm{c}}\tag{14}$$

$$\mathbf{a} = \boldsymbol{\Omega} \text{ or } \mathbf{R}^2 \mathbf{T}\_{\mathbf{c}}^{2.5} / \mathbf{P}\_{\mathbf{c}} \tag{15}$$

where Ω*a* = 0.42747 and Ωb = 0.08664. Replacing the molar volume (V) in Eq. (13) with (ZRT/P) and rearranging give.

$$\left(\mathbf{Z}^3 - \mathbf{Z}^2 + \left(\mathbf{A} - \mathbf{B} - \mathbf{B}^2\right)\mathbf{Z} - \mathbf{A}\mathbf{B} = \mathbf{0}\tag{16}$$

where

p þ

p þ

P EXP <sup>a</sup>

a VM2 � �

*a TVM*ð Þ *VM* � *<sup>b</sup>* � *<sup>c</sup>*

� �

Onnes in 1901 [22] proposed the virial equation of state as follows [23]:

*PVM* ¼ *RT* 1 þ

volumetric properties except in the critical region [24]:

<sup>1</sup> � *<sup>c</sup> VMT*<sup>3</sup> � �

<sup>2</sup> þ

*<sup>P</sup>* <sup>¼</sup> *RT VM*<sup>2</sup>

Benedict-Webb-Rubin (BWR) equation [6]:

*BoRT* � *Ao* � *Co=T*<sup>2</sup> *VM*

p þ

� � � �

attraction term in a different manner [6]:

*Inverse Heat Conduction and Heat Exchangers*

*P* þ

following equation:

attraction term:

*<sup>P</sup>* <sup>¼</sup> *RT VM* þ

**148**

a T Vð Þ <sup>M</sup> þ c

> a TVM 2

> > VM*RT*

Lorentz in 1881 [19] addressed the molecular volume term [20]:

� �

" #

2

Dieterici in 1899 [18] handled the temperature dependence of the molecular

VM � bVM VM þ b

Wohl in 1927 [21] considered the effect of temperature on the molecular

*T*2 *VM* 3

The constants a, b, and c in the equations above have different values for different substances. Several investigations proposed virial-type of EOS. Kammerlingh-

> *B VM* þ *C VM*

where B and C are not constants which are functions of temperature and are called the second and third virial coefficients. Beattie and Bridgeman in 1927 published a five-constant equation that gives a satisfactory representation of

*VM* <sup>þ</sup> *Bo* <sup>1</sup> � *<sup>b</sup>*

Benedict et al. [6] suggested a multiparameter equation of state known as the

*bRT* � *a VM* <sup>3</sup> þ

This equation may be considered a modification of the Beattie-Bridgeman equation of state where A0, B0, C0, a, b, c, α, and γ are eight adjustable parameters.

� � � �

*aα VM* <sup>6</sup> þ

� �

<sup>2</sup> þ *::* …

*VM*

*c T*2 *VM*

� �

The addition of a fourth constant (c) enabled better agreement with data. However, mathematical manipulations required in thermodynamic calculations were more difficult. So Berthelot in 1899 [17] removed the constant (c), resulting in the

ð Þ¼ V**<sup>M</sup>** � b RT (5)

ð Þ¼ V**<sup>M</sup>** � b RT (6)

ð Þ¼ V**<sup>M</sup>** � b RT (7)

¼ RT*:* (8)

ð Þ¼ *VM* � *b RT* (9)

� *Ao*ð Þ <sup>1</sup>�*a=VM VM*

<sup>3</sup> <sup>1</sup> <sup>þ</sup> *<sup>γ</sup> VM* 2

� �

(10)

(11)

(12)

*EXP* �*<sup>γ</sup> VM* 2 � �

$$B = \frac{bp}{RT} \tag{17}$$

$$A = \frac{ap}{\text{R}^2 \text{T}^{2.5}}\tag{18}$$

Three real roots in the two-phase region are yielded. The largest root corresponds to the compressibility factor of the gas phase, Zv, while the smallest positive root corresponded to that of the liquid, ZL [11].

*For mixtures*, the equation parameters a and b are usually expressed as am and bm for a hydrocarbon liquid mixture with a composition of xi:

$$a\_m = \left[\sum\_{\mathbf{i}=1}^n \mathbf{X}\mathbf{i}\,\sqrt{a\_{\mathbf{i}}}\right]^2 \tag{19}$$

$$b\_m = \sum\_{i=1}^{n} [X\_i b\_i] \tag{20}$$

am and bm for a hydrocarbon gas mixture with a composition of yi:

$$a\_m = \left[\sum\_{i=1}^n \mathcal{y}\_i \sqrt{a\_i}\right]^2 \tag{21}$$

$$b\_m = \sum\_{i=1}^{n} \left[ y\_i b\_i \right] \tag{22}$$

where n: number of components in the mixture; ai: Redlich-Kwong a parameter for the i component; bi: Redlich-Kwong b parameter for the i component; bm:

parameter b for mixture; xi: mole fraction of component i in the liquid phase; yi: mole fraction of component i in the gas phase.

Replacing the molar volume (V) in Eq. (13) with (ZRT/P) and rearranging give.

$$\left(\mathbf{Z}^3 - \mathbf{Z}^2 + \left(\mathbf{A} - \mathbf{B} - \mathbf{B}^2\right)\mathbf{Z} - \mathbf{A}\mathbf{B} = \mathbf{0}\tag{23}$$

where

$$B = \frac{b\_m p}{\text{RT}}\tag{24}$$

For pure substances the equation parameters *a* and *b* are usually expressed as.

Tc 2

In general, most EOS inputs are only the critical properties, and a centric factor

Replacing the molar volume (V) in the equation with (ZRT/p) and rearranging

*<sup>B</sup>* <sup>¼</sup> *bmp*

*<sup>A</sup>* <sup>¼</sup> *amP*

for a hydrocarbon liquid mixture with a composition of xi:

*am* <sup>¼</sup> <sup>X</sup> *i*

*am* <sup>¼</sup> <sup>X</sup> *i*

X *j*

*xixj*

*bm* <sup>¼</sup> <sup>X</sup> *i*

The following is the calculation for am and bm for a gas mixture with a

*bm* <sup>¼</sup> <sup>X</sup> *i*

in the "a" parameter expression to provide more flexibility to the EOS and designed to characterize any binary system formed by components i and j in the hydrocarbon mixture [32]. Vidal and Daubert [33], Graboski and Daubert [34], and Slot-Petersen [35] suggested that no BIs were required for hydrocarbon systems. However, with no hydrocarbons present, binary interaction parameters can improve the phase in volumetric behavior predictions of the mixture by the SRK EOS for compressibility factor calculations of the gas or the liquid phases [34, 36, 37]. The equilibrium ratio, Ki, that is, Ki = yi /xi, can be redefined in terms of the

A binary interaction parameter (BI), classically noted as kij, is usually involved

X *j yi yj*

*For mixtures*, the equation parameters a and b are usually expressed as am and bm

Z3 � <sup>Z</sup><sup>2</sup> <sup>þ</sup> <sup>A</sup> � <sup>B</sup> � <sup>B</sup><sup>2</sup> � � <sup>Z</sup> � AB <sup>¼</sup> <sup>0</sup> (31)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *aiajαiα<sup>j</sup>* <sup>1</sup> � *kij* <sup>q</sup> � � �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *aiajαiα<sup>j</sup>* <sup>1</sup> � *kij* <sup>q</sup> � � �

where Ωa and Ωb are the SRK dimensionless pure component parameters:

<sup>a</sup> <sup>¼</sup> <sup>Ω</sup>a R<sup>2</sup>

of each components is shown in **Table 1**.

*DOI: http://dx.doi.org/10.5772/intechopen.89919*

give the compressibility factor Z:

Ωa = 0.42747. Ωb = 0.08664.

*Equation of State*

where

composition of yi:

fugacity of component:

**151**

b ¼ ΩbRTc*=*Pc (29)

*=*Pc (30)

RT (32)

ð Þ RT <sup>2</sup> (33)

½ � *Xi bi* (35)

½ � *yi bi* (37)

(34)

(36)

$$A = \frac{a\_m P}{\mathbf{R}^2 \mathbf{T}^{2.5}}\tag{25}$$

Then the compressibility factor of the gas phase or the liquid can be calculated.

Joffe and Zudkevitch [26] showed that a substantial improvement in the representation of fugacity of gas mixtures could be obtained by treating interaction parameters as empirical parameters [26]. Spear et al. [27] also states that the RK equation of state could be used to calculate the vapor-liquid critical properties of binary mixtures [28]. Chueh and Prausnitz [29] showed that the RK equation can be adapted to predict both vapor and liquid properties. Spear et al. [28] gave seven examples of systems for which the vapor-liquid critical properties of hydrocarbon mixtures could be calculated by using the RK equation of state. Carnahan and Starling [30] used the Redlich-Kwong equation of state to calculate the gas-phase enthalpies for a variety of substances [30]. Their results showed that the Redlich-Kwong equation was a significant improvement over the Van der Waals equation. Other workers applied the Redlich-Kwong equation to the critical properties and the high-pressure phase equilibria of binary mixtures. The results showed that the accuracy of the Redlich-Kwong equation of state calculations for ternary systems was only slightly less than that for the constituent binaries [31].

The success of the Redlich-Kwong equation has been the impetus for many further empirical improvements. One of the milestones in developing of CEOS was reported by Soave [4]. His development in the evaluation of the parameter in the attraction pressure term for the RK equation is shown in (Eq. (22)). Soave replaced the term (a/T0.5) in Eq. (22) with a more general temperature-dependent term, denoted by a α (T), to give

$$
\rho = \frac{RT}{V - b} - \frac{a \; a(T)}{V(V + b)} \tag{26}
$$

where α(T) is a dimensionless factor. Soave used vapor pressures of pure components to introduce an expression for the temperature correction parameter α(T). At temperatures other than the critical temperature, the correction parameter α(T) was defined by the following equation:

$$a(T) = \left[1 + m\left(1 - \sqrt{T\_{\mathbf{r}}}\right)\right]^2\tag{27}$$

Soave correlated the parameter "m" with the centric factor (ω) to give.

$$\mathbf{m} = \mathbf{0}.480 + \mathbf{1.574}\mathbf{w}, \ -\mathbf{0.176}\mathbf{w}^2 \tag{28}$$

where Tr: reduced temperature, °R; ω: a centric factor of the substance; T: system temperature, °R.

For pure substances the equation parameters *a* and *b* are usually expressed as.

$$\mathbf{b} = \mathbf{Q} \mathbf{b} \, \mathbf{R} \, \mathbf{T}\_{\mathbf{c}} / \mathbf{P}\_{\mathbf{c}} \tag{29}$$

$$\mathbf{a} = \mathbf{\Omega} \mathbf{a} \, \mathbf{R}^2 \mathbf{T}\_{\mathbf{c}} \, ^2 / \mathbf{P}\_{\mathbf{c}} \, \tag{30}$$

In general, most EOS inputs are only the critical properties, and a centric factor of each components is shown in **Table 1**.

where Ωa and Ωb are the SRK dimensionless pure component parameters: Ωa = 0.42747.

Ωb = 0.08664.

Replacing the molar volume (V) in the equation with (ZRT/p) and rearranging give the compressibility factor Z:

$$\left(\mathbf{Z}^3 - \mathbf{Z}^2 + \left(\mathbf{A} - \mathbf{B} - \mathbf{B}^2\right)\mathbf{Z} - \mathbf{A}\mathbf{B} = \mathbf{0}\tag{31}$$

where

parameter b for mixture; xi: mole fraction of component i in the liquid phase; yi:

Replacing the molar volume (V) in Eq. (13) with (ZRT/P) and rearranging give.

*<sup>B</sup>* <sup>¼</sup> *bmp*

*<sup>A</sup>* <sup>¼</sup> *amP* R2

sentation of fugacity of gas mixtures could be obtained by treating interaction parameters as empirical parameters [26]. Spear et al. [27] also states that the RK equation of state could be used to calculate the vapor-liquid critical properties of binary mixtures [28]. Chueh and Prausnitz [29] showed that the RK equation can be adapted to predict both vapor and liquid properties. Spear et al. [28] gave seven examples of systems for which the vapor-liquid critical properties of hydrocarbon mixtures could be calculated by using the RK equation of state. Carnahan and Starling [30] used the Redlich-Kwong equation of state to calculate the gas-phase enthalpies for a variety of substances [30]. Their results showed that the Redlich-Kwong equation was a significant improvement over the Van der Waals equation. Other workers applied the Redlich-Kwong equation to the critical properties and the high-pressure phase equilibria of binary mixtures. The results showed that the accuracy of the Redlich-Kwong equation of state calculations for ternary systems

was only slightly less than that for the constituent binaries [31].

*<sup>ρ</sup>* <sup>¼</sup> *RT*

denoted by a α (T), to give

T: system temperature, °R.

**150**

was defined by the following equation:

The success of the Redlich-Kwong equation has been the impetus for many further empirical improvements. One of the milestones in developing of CEOS was reported by Soave [4]. His development in the evaluation of the parameter in the attraction pressure term for the RK equation is shown in (Eq. (22)). Soave replaced the term (a/T0.5) in Eq. (22) with a more general temperature-dependent term,

*<sup>V</sup>* � *<sup>b</sup>* � *<sup>a</sup> <sup>α</sup>*ð Þ *<sup>T</sup>*

where α(T) is a dimensionless factor. Soave used vapor pressures of pure components to introduce an expression for the temperature correction parameter α(T). At temperatures other than the critical temperature, the correction parameter α(T)

h i � � p <sup>2</sup>

*T*r

<sup>m</sup> <sup>¼</sup> <sup>0</sup>*:*<sup>480</sup> <sup>þ</sup> <sup>1</sup>*:*574ϖ, � <sup>0</sup>*:*176ϖ<sup>2</sup> (28)

*<sup>α</sup>*ð Þ¼ *<sup>T</sup>* <sup>1</sup> <sup>þ</sup> *<sup>m</sup>* <sup>1</sup> � ffiffiffiffiffi

Soave correlated the parameter "m" with the centric factor (ω) to give.

where Tr: reduced temperature, °R; ω: a centric factor of the substance;

Then the compressibility factor of the gas phase or the liquid can be calculated. Joffe and Zudkevitch [26] showed that a substantial improvement in the repre-

Z3 � <sup>Z</sup><sup>2</sup> <sup>þ</sup> <sup>A</sup> � <sup>B</sup> � <sup>B</sup><sup>2</sup> � � <sup>Z</sup> � AB <sup>¼</sup> <sup>0</sup> (23)

RT (24)

<sup>T</sup>2*:*<sup>5</sup> (25)

*V V*ð Þ <sup>þ</sup> *<sup>b</sup>* (26)

(27)

mole fraction of component i in the gas phase.

*Inverse Heat Conduction and Heat Exchangers*

where

$$B = \frac{b\_m p}{\text{RT}}\tag{32}$$

$$A = \frac{a\_m P}{\left(\text{RT}\right)^2} \tag{33}$$

*For mixtures*, the equation parameters a and b are usually expressed as am and bm for a hydrocarbon liquid mixture with a composition of xi:

$$a\_m = \sum\_{i} \sum\_{j} \left[ \mathbf{x}\_i \mathbf{x}\_j \sqrt{a\_i a\_j \alpha\_i a\_j \left(\mathbf{1} - k\_{ij}\right)} \right] \tag{34}$$

$$b\_m = \sum\_i [\mathbf{X}i\ bi] \tag{35}$$

The following is the calculation for am and bm for a gas mixture with a composition of yi:

$$\mathfrak{a}\_{m} = \sum\_{i} \sum\_{j} \left[ \mathcal{y}\_{i} \mathcal{y}\_{j} \sqrt{a\_{i} a\_{j} a\_{i} a\_{j} \left(\mathbf{1} - k\_{ij}\right)} \right] \tag{36}$$

$$b\_m = \sum\_i [yi\,bi] \tag{37}$$

A binary interaction parameter (BI), classically noted as kij, is usually involved in the "a" parameter expression to provide more flexibility to the EOS and designed to characterize any binary system formed by components i and j in the hydrocarbon mixture [32]. Vidal and Daubert [33], Graboski and Daubert [34], and Slot-Petersen [35] suggested that no BIs were required for hydrocarbon systems. However, with no hydrocarbons present, binary interaction parameters can improve the phase in volumetric behavior predictions of the mixture by the SRK EOS for compressibility factor calculations of the gas or the liquid phases [34, 36, 37]. The equilibrium ratio, Ki, that is, Ki = yi /xi, can be redefined in terms of the fugacity of component:


**Table 1.** *Physical properties of each components.* *K i* ¼

<sup>v</sup> = fugacity of component

*DOI: http://dx.doi.org/10.5772/intechopen.89919*

*ΦLi* = fugacity coefficient of component

*ψj* ¼ X*j xj*

Fugacity coefficient of component i in the gas phase:

� ln *Z i* � *B* � �

*ψj* ¼ X*j yj*

*a m* ¼ X*i*

X*j x i xj*

X*j y iyj*

�

�

� *AB* � � 2 *ψ i a m* � *b i b m* � � ln 1

Heat exchanger is an energy (heat) exchange equipment, where it transfers the heat from a working medium to another working medium. Knowing heat exchanger is important in wildly fields as in aerospace, petrochemical industry, refrigeration, and other fields. The optimization design of the heat exchanger is a great significance to industry process to reduce production cost, realize energy conservation, and reduce energy consumption [38]. The development technique for different types of the heat exchanger has been reviewed by many researchers. The development method can be by two ways: passive method and active method. The passive method is to generate swirling flow and disturb the thermal boundary layer by installing vortex generator or tabulators such as baffle, rib, winglet, wing, etc.

*a m* ¼ X*i*

where fi

*Equation of State*

*f Li XiP* !

where

ln *ϕj* � � ¼ *b i Z i* � 1 � � *bm*

where:

**3. Heat exchanger**

**153**

nent i in the liquid phase:

¼ *ln*

*ΦLi* � � ¼ *b i Z l* � *1* � � *bm*

phase;

*ln*

*f Li <sup>=</sup>*ð Þ *XiP* h i

> " i

*ln Z<sup>L</sup>*

*<sup>f</sup> Vi <sup>=</sup> yiP* � � h i

nent "i" in the liquid phase; *Φvi* = fugacity coefficient of component "i" in the vapor

Soave proposed the following expression for the fugacity coefficient of compo-

� *B* � �

¼ *ΦLi Φvi*

" in the gas phase; fi

� *AP* � �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *aiajαiα<sup>j</sup>* <sup>1</sup> � *kij* <sup>q</sup> � �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *aiajαiα<sup>j</sup>* <sup>1</sup> � *kij* <sup>q</sup> � �

" in the liquid phase.

2 *ψ i a m* � *b i b m* � � *ln 1*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *aiajαiα<sup>j</sup>* <sup>1</sup> � *kij* <sup>q</sup> � � � � (40)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *aiajαiα<sup>j</sup>* <sup>1</sup> � *kij* <sup>q</sup> � � � � (43)

þ *BZi* � � (42)

" i (38)

þ *BZL* � �

(39)

(41)

(44)

<sup>L</sup> = fugacity of compo-

*Inverse Heat Conduction and Heat Exchangers*

**152**

*Equation of State DOI: http://dx.doi.org/10.5772/intechopen.89919*

$$K\_i = \frac{\left[f\_i^L / (\text{Xi}P)\right]}{\left[f\_i^V / (\text{y}\_i P)\right]} = \frac{\Phi\_i^L}{\Phi\_i^v} \tag{38}$$

where fi <sup>v</sup> = fugacity of component "i" in the gas phase; fi <sup>L</sup> = fugacity of component "i" in the liquid phase; *Φ<sup>v</sup> <sup>i</sup>* = fugacity coefficient of component "i" in the vapor phase; *Φ<sup>L</sup> <sup>i</sup>* = fugacity coefficient of component "i" in the liquid phase.

Soave proposed the following expression for the fugacity coefficient of component i in the liquid phase:

$$\ln\left(\frac{f\_i^L}{X\_i P}\right) = \ln\left(\Phi\_i^L\right) = \frac{b\_i \left(Z^l - 1\right)}{b\_m} \ln\left(Z^L - B\right) - \left(\frac{A}{P}\right) \left[\frac{2\nu\_i}{a\_m} - \frac{b\_i}{b\_m}\right] \ln\left[1 + \frac{B}{Z^L}\right] \tag{39}$$

where

$$\nu\_{j} = \sum\_{j} \left[ \mathbf{x}\_{j} \sqrt{a\_{i} a\_{j} a\_{i} a\_{j} \left(\mathbf{1} - k\_{ij}\right)} \right] \tag{40}$$

$$a\_m = \sum\_{i} \sum\_{j} \left[ \mathbf{x}\_i \mathbf{x}\_j \sqrt{a\_i a\_j \alpha\_i a\_j \left(\mathbf{1} - k\_{ij}\right)} \right] \tag{41}$$

Fugacity coefficient of component i in the gas phase:

$$\ln\left(\phi\_{j}\right) = \frac{b\_{i}\left(Z^{i} - 1\right)}{b\_{m}} - \ln\left(Z^{i} - B\right) - \left(\frac{A}{B}\right)\left[\frac{2\mu\_{i}}{a\_{m}} - \frac{b\_{i}}{b\_{m}}\right]\ln\left[1 + \frac{B}{Z^{i}}\right] \tag{42}$$

where:

$$\nu\_{j} = \sum\_{j} \left[ \nu\_{j} \sqrt{a\_{i} a\_{j} a\_{i} a\_{j} \left( 1 - k\_{ij} \right)} \right] \tag{43}$$

$$a\_m = \sum\_{i} \sum\_{j} \left[ \mathbf{y}\_j \mathbf{y}\_j \sqrt{a\_i a\_j a\_i a\_j \left(\mathbf{1} - k\_{ij}\right)} \right] \tag{44}$$
