Physical properties of each components.
Abstract
An equation of state (EOS) is a thermodynamic expression that relates pressure (P), temperature (T), and volume (V). This equation is used to describe the state of reservoir fluids at given conditions. The cubic equations of state (CEOS) such as Van der Waals, RedlichKwong, Soave, and PengRobinson are simple models that have been widely used in the oil industry. This chapter expressed literature for EOS that varies from simple expressions to multiple constant and convoluted types of equations. Many attempts have been made to describe the thermodynamic behavior of fluids to predict their physical properties at given conditions. So, several forms of the equation of state have been presented to the oil industry in order to calculate reservoir fluid properties. The 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.
Keywords
 EOS types
 EOS importance
 EOS history
 cubic equation of state
 heat transfer
1. Introduction
In the last six decades, petroleum engineers realized the importance of using EOS for PVT modeling in addition to the following [1]:
Predicting physical properties of the black oil in the laboratories is very expensive.
Difficulty to obtain a representative sample due to reservoir nature or operation issues.
Insufficient sample volume in order to obtain a complete analysis.
Error in lab analyses.
Samples are nonrepresentative (not monophasic sample, OBM contamination, etc.).
Lab report quality check.
Estimating the hydrocarbon reserves.
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], SoaveRedlichKwong [4], and PengRobinson [5] equations give reasonable results for the thermodynamic behavior of real fluids.
1.1.2 Second class of EOS
These EOS are noncubic 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 firstclass EOS is more useful because it provides an analytical solution than the more complex and complicated noncubic second type and nonanalytical third type that require timeconsuming iterative calculations. In general, the overall performance in fluid properties prediction is somewhat better using the SoaveRedlichKwong (SRK) equation than using the RedlichKwong (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^{−1}). 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].
where P: absolute pressure, psia; V: volume, ft^{3}; T: absolute temperature, °R; R: the universal gas constant (10.73159 ft^{3} psia °R^{−1} lbmole^{−1}; n: number of moles of gas, lbmole.
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 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]:
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:
The gas molecule volume is very small compared to the volume of the container.
There are no attractive or repulsive forces between the gas molecules or the walls of the container.
Van der Waals attempted to eliminate these assumptions in the development of an empirical EOS for the real gases.
Van der Waals introduced the following equation (Eq. (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 covolume 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, p_{repulsion}, is represented by the term RT/(V_{m} – b), and the attraction pressure, p_{attraction}, is described by a/V_{m} ^{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/V_{m}M^{2}) was proposed. Clausius in 1880 [15] proposed that the molecular attraction term was inversely proportional to temperature [16]:
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:
Dieterici in 1899 [18] handled the temperature dependence of the molecular attraction term in a different manner [6]:
Lorentz in 1881 [19] addressed the molecular volume term [20]:
Wohl in 1927 [21] considered the effect of temperature on the molecular attraction term:
The constants a, b, and c in the equations above have different values for different substances. Several investigations proposed virialtype of EOS. KammerlinghOnnes in 1901 [22] proposed the virial equation of state as follows [23]:
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 fiveconstant equation that gives a satisfactory representation of volumetric properties except in the critical region [24]:
Benedict et al. [6] suggested a multiparameter equation of state known as the BenedictWebbRubin (BWR) equation [6]:
This equation may be considered a modification of the BeattieBridgeman equation of state where A_{0}, B_{0}, C_{0}, 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 RedlichKwong (RK) (1949) which is demonstrated by an adjustment of the Van der Waals’s attraction term (a/V_{m} ^{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 temperaturedependent term (Eq. (13)) [3]:
where Ω
Replacing the molar volume (V) in Eq. (13) with (ZRT/P) and rearranging give.
where
Three real roots in the twophase region are yielded. The largest root corresponds to the compressibility factor of the gas phase, Z_{v}, while the smallest positive root corresponded to that of the liquid, Z_{L} [11].
a_{m} and b_{m} for a hydrocarbon gas mixture with a composition of y_{i}:
where n: number of components in the mixture; a_{i}: RedlichKwong a parameter for the i component; b_{i}: RedlichKwong b parameter for the i component; b_{m}: parameter b for mixture; x_{i}: mole fraction of component i in the liquid phase; y_{i}: mole fraction of component i in the gas phase.
Replacing the molar volume (V) in Eq. (13) with (ZRT/P) and rearranging give.
where
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 vaporliquid 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 vaporliquid critical properties of hydrocarbon mixtures could be calculated by using the RK equation of state. Carnahan and Starling [30] used the RedlichKwong equation of state to calculate the gasphase enthalpies for a variety of substances [30]. Their results showed that the RedlichKwong equation was a significant improvement over the Van der Waals equation. Other workers applied the RedlichKwong equation to the critical properties and the highpressure phase equilibria of binary mixtures. The results showed that the accuracy of the RedlichKwong equation of state calculations for ternary systems was only slightly less than that for the constituent binaries [31].
The success of the RedlichKwong 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/T^{0.5}) in Eq. (22) with a more general temperaturedependent term, denoted by a α (T), to give
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:
Soave correlated the parameter “m” with the centric factor (ω) to give.
where T_{r}: reduced temperature, °R; ω: a centric factor of the substance; T: system temperature, °R.
For pure substances the equation parameters
In general, most EOS inputs are only the critical properties, and a centric factor of each components is shown in Table 1.
Compound  Formula  Molecular weight  Critical pressure (psla)  Critical temperature (°F)  Critical volume (ft^{3}/lb)  Liquid specific gravity (water = 1)  Gas specific (air = 1)  Acentric factor 

Methane  CH_{4}  16.042  667.0  −116.66  0.0985  (0.3)  0.55400  0.0115 
Ethane  C_{2}H_{6}  30.069  706.6  89.92  0.0775  0.35643  1.03830  0.0994 
Propane  C_{3}H_{8}  44.096  615.5  205.92  0.0728  0.50738  1.52270  0.1529 
Isobutane  C_{4}H_{10}  58.122  527.9  274.41  0.0715  0.56295  2.00710  0.1865 

C_{4}H_{10}  58.122  550.9  305.55  0.0703  0.58408  2.00710  0.2003 
Isopentane  C_{5}H_{12}  72.149  490.4  369  0.0685  0.62460  2.49140  0.2284 

C_{5}H_{12}  72.149  488.8  385.8  0.0676  0.63113  2.49140  0.2515 

C_{6}H_{14}  86.175  436.9  453.3  0.0688  0.66404  2.97580  0.2993 

C_{7}H_{16}  100.202  396.8  512.9  0.0682  0.68819  3.46020  0.3483 

C_{8}H_{18}  114.229  360.7  564.2  0.0673  0.70698  3.94450  0.3977 

C_{9}H_{20}  128.255  330.7  610.8  0.693  0.72186  4.42890  0.4421 

C_{10}H_{22}  142.282  304.6  652.2  0.0703  0.73406  4.91330  0.4875 
Carbon monoxide  CO  28.01  506.7  −220.63  0.0527  0.79265  0.96720  0.0510 
Carbon dioxide  CO_{2}  44.01  1070.0  87.76  0.0343  0.82203  1.51970  0.2239 
Hydrogen sulfide  H_{2}S  34.082  1306.5  212.81  0.0462  0.80269  1.17690  0.1010 
Air  —  28.9586  551.9  −220.97  0.0458  0.87603  1.00000  — 
Hydrogen  H_{2}  2.0159  190.7  −399.9  0.5319  0.07087  0.06961  −0.2140 
Oxygen  O_{2}  31.9988  731.4  −181.43  0.0367  1.14230  1.10500  0.0222 
Nitrogen  N_{2}  28.0135  492.5  −232.53  0.0511  0.80687  0.96740  0.0372 
Water  H_{2}O  18.0153  3200.1  705.1  0.04975  1.00000  0.62210  0.3443 
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:
where
The following is the calculation for a_{m} and b_{m} for a gas mixture with a composition of y_{i}:
A binary interaction parameter (BI), classically noted as k_{ij}, 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 SlotPetersen [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, K_{i}, that is, K_{i} = yi /xi, can be redefined in terms of the fugacity of component:
where f_{i}
^{v} = fugacity of component “i” in the gas phase; f_{i}
^{L} = fugacity of component “i” in the liquid phase;
Soave proposed the following expression for the fugacity coefficient of component i in the liquid phase:
where
Fugacity coefficient of component i in the gas phase:
where:
3. Heat exchanger
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. The active method is to add the external power to increase efficiency and heat transfer rate such as vibration. So the use of the active method must consider both benefit of the system and additional power cost [39].
3.1 Optimization techniques of heat exchanger
The optimization techniques of heat exchanger can be shown at three different stages as the following [40]:
Identification of the lowest initial cost of a heat exchanger design that meets the process specifications.
Identification of a heat exchanger design that will work most acceptably over the plant lifetime.
Identification of the minimum total cost of the process by choosing heat exchangers system and auxiliary components that will make the best plant process specifications.
3.2 Criteria of the ideal heat exchanger
The dissimilarities of optimization techniques levels can be understood if we list the required criteria of the ideal heat exchanger as follows [41]:
The heat exchangers should be flexible enough to meet specifications process such as normal fouling transients and seasonal and diurnal changes in service stream temperatures.
Special requirements as weight, length, or inventory standards are important for heat exchangers especially in retrofit applications.
The heat exchanger must endure operation under standard and foreseeable operating conditions, maintain the mechanical stresses of manufacturing transport, and minimize the effects of fouling and corrosion.
The heat exchanger must be maintainable, cleaning, repair or replacement and its components as gaskets and tubes with minimum downtime.
The exchanger must achieve process specifications, i.e., achieve any changes in the thermal conditions by allowable pressure drops.
There are other requirements, such as experience, capability of operating, maintenance personnel, and manufacturing time.
The exchanger should cost as little as possible.
4. Fluid flow and heat transfer mechanism
The heat transfer and fluid flow mechanism in porous media is important in engineering and industrial fields such as petroleum and chemical engineering [42]. This mechanism occurs for many studies such as in microchannels and nanofluids.
Example 1: In case of study characteristics of fluid flow and heat transfer in the (100) silicon microchannel heat sink, the heat convection capabilities in the phase changes as well as in a singlephase flow and the mechanism of bubble nucleation. In the heat transfer characteristics, the results illustrate that changing in the phase process in the microchannels reduces environment working temperature and absorbs the heat. Six different microchannel geometries are selected for the heat transfer experiment as shown in Table 2.
Chip name  Width (μm) W_{c} 
Depth (μm) H_{c} 
Hydraulic diameter (μm)  Number of channels 

Chip 1  400  260  221  10 
Chip 2  300  130  150  13 
Chip 3  250  184  134  15 
Chip 4  200  148  109  19 
Chip 5  150  113  83  25 
Chip 6  100  78  57  38 
Figure 1 shows that the decreasing wall temperature phenomenon during the phase change is the same as Peng and Wang [43].
On the aspect of fluid flow characteristics, the effects of the viscosity and friction coefficient of the fluid in the microchannels are much significant than the macros. Where the specifications of the sink are registered in Table 2, Chip 1–4 are prepared for fluid flow experiment. The friction factor is decreasing with the power of Reynolds number as shown in Figure 2 [44].
Nanoparticle types and concentration in the base fluids
Changing in thermophysical properties such as viscosity, surface tension, thermal conductivity, density, and heat capacity
The operation condition especially the mass and heat fluxes
The modification of wettability and capillary wicking force surface roughness
During the last 2 years, there were some review papers which outlined the subject of boiling heat transfer using nanofluids as a new category in thermal fluids. Available results reported that the effect of nanoparticles on the flow boiling HTC is conflicting, but the CHF could enhance on 50%. During the boiling process, parameters such as flow pattern and pressure drop were affected by the deposition of nanoparticles [46]. Authors concluded that using nanofluids might intensify the boiling HT and CHF, depending on many parameters related to additives, nanoparticles, geometry concentration, and fluid properties [47].
Their work shows how the nanofluids can achieve high heat flux with small temperature differences throughout the boiling process, which modify the critical heat flux [48]. All article reviewers said that nanofluids is a complicated phenomenon and it is not fully understood from mechanism of boiling heat transfer and twophase flow. Collected studies show enhancement in CHF, and its improvement could achieve more than 50% [49]. HTC behavior could increase or decrease during flow boiling and pool, and it depends on several parameters related to surface texture on thermophysical properties [50]. Wang et al. shows the mechanism of depositing nanoparticles on the heating surface and creating the porous layer as shown in Figure 3. The results show that the CHF of flow boiling is enhanced up to 18% as compared to conventional fluid. This enhancement increases with increasing some parameters, for example, the pressure system and the channel diameter [45].
Example 2: in the case of study fluid flow and heat transfer characteristics using nanofluid in a singlephase turbulent flow by using helically corrugated tubes, pitchtodiameter ratio (P/DH = 0.18, 0.22 and 0.27) and ribheighttodiameter ratio (e/DH = 0.02, 0.04 and 0.06) of helically corrugated tubes on the heat transfer enhancement, isothermal friction, and thermal performance factor in a concentric tube heat exchanger are examined. Results illustrate that the thermal performance of the corrugated tube and heat transfer are increased as compared to those of the smooth tube. The rate increase in heat transfer rate is between 123 and 232%, depending on the rib height/pitch ratios and Reynolds number. The friction factor (average) of the corrugated tube is between 1.46 and 1.93 over the smooth tube [51].
Abbreviations
EOS  equation of state 
BWR  BenedictWebbRubin 
RK  RedlichKwong 
SRK  SoaveRedlichKwong 
CEOS  cubic equations of state 
MSRKE  modified SoaveRedlichKwong equation of state 
p  system pressure, psia 
Pc  critical pressure, psia 
Pr  reduced pressure, psia 
pi  initial pressure, psia 
T  system temperature, oF 
Tc  critical temperature, oR 
Tr  reduced temperature, oR 
Ti  initial temperature, oF 
a  equation of state attraction parameter 
b  equation of state repulsion parameter 
B  second virial coefficient 
C  third virial coefficient 
a, b, c, A0, B0, C0  constant in BenedictWebbRubin equation 
a, b, c, A0, B0  constant in Beattie and Bridgeman equation of state 
fv i  fugacity of component i in the gas phase 
fL j  fugacity of component j in the liquid phase 
Øv i  fugacity coefficient of component i in the vapor phase 
ØL i  fugacity coefficient of component i in the liquid phase 
x  mole fraction of gas phase 
y  mole fraction of liquid phase 
k  equilibrium ratio for the component 
Kij  interaction coefficient parameter 
A, B  parameter in SoaveRedlichKwong equation of state 
aT  temperaturedependent coefficient in SoaveRedlichKwong equation of state 
ac  constant coefficient in SoaveRedlichKwong equation of state 
aTi  temperaturedependent coefficient of component i 
m  parameter in SoaveRedlichKwong equation of state 
Z  compressibility factor 
Zi  the mole fraction of component in the mixture 
ZL  compressibility factor of the liquid phase 
Zv  compressibility factor of the gas phase 
ω  a centric factor of the substance 
nl  number of moles in liquid phase 
nv  number of moles in gas phase 
nt  total number of moles of mixture 
Fl  one mole liquid of total fraction 
Fv  one mole vapor of total fraction 
Xi  mole fraction in the liquid phase 
yi  mole fraction in the vapor phase 
γ  constant in BenedictWebbRubin equation of state 
References
 1.
Asadisaghandi J, Tahmasebi P. Comparative evaluation of backpropagation neural network learning algorithms and empirical correlations for prediction of oil PVT properties in Iran oilfields. Journal of Petroleum Science and Engineering. 2011; 78 (2):464475  2.
Van der Waals JD. Over de Continuiteit van den Gasen Vloeistoftoestand. Leiden: A.W. Sijthoff; 1873  3.
Redlich O, Kwong JN. On the thermodynamics of solutions. V. An equation of state. Fugacities of gaseous solutions. Chemical Reviews. 1949; 44 (1):233244  4.
Soave G. Equilibrium constants from a modified RedlichKwong equation of state. Journal of Chemical Engineering Science. 1972; 27 :11971203  5.
Peng DY, Robinson DB. A new twoconstant equation of state. Journal of Engineering Chemical Fundamental. 1976; 15 :5964  6.
Benedict M, Webb GB, Rubin LC. An empirical equation for thermodynamic properties of light hydrocarbons and their mixtures I. Methane, ethane, propane and nbutane. The Journal of Chemical Physics. 1940; 8 (4):334345  7.
Goodwin RD. Provisional Thermodynamic Functions of Propane, from 85 to 700 K at Pressures to 700 Bar. National STANDARD Reference Data System. Institute for Basic Standards. 1977  8.
Patel NC, Teja AS. A new cubic equation of state for fluids and fluid mixtures. Chemical Engineering Science. 1982; 37 (3):463473  9.
Whitson CH, Brulé MR. Phase Behavior, Henry L. Doherty Memorial Fund of AIME, Society of Petroleum Engineers; 2000  10.
Atllhan M. A new cubic equation of state [M.Sc. thesis]. Texas: A&M University; 2004  11.
Ahmed T. Equations of State and PVT Analysis. Gulf Professional Publishing; 2013  12.
Mansour E, Desouky S, Batanoni M, Mahmoud M, Farag A, ElDars F. Modification proposed for SRK equation of state. Oil and Gas Journal. 2012; 110 (6):7891  13.
Yao Y, Xie T, Gao Y. Physical Chemistry Handbook. Shanghai Science and Technology Publishing Company; 1985. p. 706  14.
Wei YS, Sadus RJ. Equations of state for the calculation of fluid phase equilibria. AIChE Journal. 2000; 46 :169196  15.
Clausius R. Ueber des verhalten der kohlensaure in begug auf druck, volumen and temperatur. Annual Physical Chemistry. 1880; 9 :337  16.
William D. The Properties of Petroleum Fluids Handbook. Vol. 148. Tulsa: PennWell Publishing Company; 1994  17.
Berthelot DJ. Sur une méthode purement physique pour la détermination des poids moléculaires des gaz et des poids atomiques de leurs éléments. Journal of Physics. 1899; 8 :263  18.
Dieterici C. Ueber den kritischen Zustand. Annual Physical Chemistry. 1899; 30 :685  19.
Lorentz. Wied. Annual Physical Chemistry. 1881; 12 (127):660  20.
Boyd MF. Studies on the bionomics of north American AnopheLines: Physical and chemical factors in their relation to the distribution of larvae in northeastern North Carolina. American Journal of Hygiene. 1929; 9 (2):6  21.
Wohl KZ. Physical Chemistry. 1927; 133 :305  22.
KammerlinghOnnes HK. Commununications from the Physics Laboratory. Vol. 71. Leiden, Holland; 1901  23.
Adamson AW, Gast AP. Physical Chemistry of Surfaces. New York: Interscience; 1967  24.
Beattie JA, Bridgeman OC. A new equation of state for fluids. I. Application to gaseous ethyl ether and carbon dioxide1. Journal of the American Chemical Society. 1927; 49 (7):16651667  25.
Mansour E, Farag A, ElDars F, Desouky S, Batanoni M, Mahmoud M. Predicting PVT properties of Egyptian crude oils by a modified SoaveRedlichKowng equation of state. Egyptian Journal of Petroleum. 2013; 22 (1):137148  26.
Joffe J, Zudkevitch D. Fugacity coefficients in gas mixtures containing light hydrocarbons and carbon dioxide. Industrial & Engineering Chemistry Fundamentals. 1966; 5 (4):455462  27.
Spear RR, Robinson RL, Chao KC. Critical states of mixtures and equations of state. Industrial Engineers Chemical Fundamental. 1969; 8 :28  28.
Spear RR, Robinson RL Jr, Chao KC. Critical states of ternary mixtures and equations of state. Industrial & Engineering Chemistry Fundamentals. 1971; 10 (4):588592  29.
Chueh PL, Prausnitz JM. Vapourliquid equilibria at high pressures: Calculation of partial molar volumes in nonpolar liquid mixtures. AIChE Journal. 1967; 13 :1099  30.
Carnahan NF, Starling KE. Intermolecular repulsions and the equation of state for fluids. AICHE Journal. 1972; 18 (6):11841189  31.
Mansour E, AlSabagh A, Desouky S, Zawawy F, Ramzi M. Experimental approach of minimum miscibility pressure for CO_{2} miscible flooding: Application to Egyptian oil fields. International Journal of New Technology and Research. 2016; 2 (5):8  32.
Mansour E, AlSabagh A, Desouky S, Zawawy F, Ramzi M. A laboratory investigation of carbon dioxideenhanced oil recovery by focusing on CO_{2}oil physical properties. Egyptian Journal of Petroleum. 2019; 28 (1):2126  33.
Vidal J, Daubert T. Equations of statereworking the old forms. Journal of Chemical Engineering Society. 1978; 33 :787791  34.
Graboski MS, Daubert TE. A modified soave equation of state for phase equilibrium calculations. 1. Hydrocarbon systems. Industrial & Engineering Chemistry Process Design and Development. 1978; 17 (4):443448  35.
SlotPetersen C. A systematic and consistent approach to determine binary interaction coefficients for the PengRobinson equation of state. In: 62nd, Annual Technical Conference of the SPE, Dallas, TX; 2730 September 1987  36.
SlotPetersen C. A systematic and consistent approach to determine binary interaction coefficients for the PengRobinson equation of state (includes associated papers 20308 and 20393). SPE Reservoir Engineering. 1989; 4 (04):488494  37.
Vidal J. Equations of statereworking the old forms. Fluid Phase Equilibria. 1983; 13 :1533  38.
Zeugmann T, Poupart P, Kennedy J, Jin X, Han J, Saitta L, et al. Particle Swarm Optimization. Boston, MA: Springer US; 2011. pp. 760766  39.
Boonloi A, Jedsadaratanachai W. Numerical study on flow and heat transfer mechanisms in the heat exchanger channel with Vorifice at various blockage ratios, gap spacing ratios, and flow directions. Modelling and Simulation in Engineering. 2019; 2019 :5  40.
Rao RV, Patel V. Multiobjective optimization of heat exchangers using a modified teachinglearningbased optimization algorithm. Applied Mathematical Modelling. 2013; 37 (3):11471162  41.
Yao J. A review of industrial heat exchange optimization. In: IOP Conference Series: Earth and Environmental Science. IOP Publishing; 2018  42.
Chatti S, Ghabi C, Mhimid A. Fluid flow and heat transfer in porous media and post heated obstacle: Lattice Boltzmann simulation. International Journal of Heat and Technology. 2016; 34 (3):377385  43.
Peng X, Peterson G, Wang B. Frictional flow characteristics of water flowing through rectangular microchannels. Experimental Heat Transfer An International Journal. 1994; 7 (4):249264  44.
Chen YT, Kang SW, Tuh WC, Hsiao TH. Experimental investigation of fluid flow and heat transfer in microchannels. 淡江理工學刊. 2004; 7 (1):1116  45.
Kamel MS, Lezsovits F, Hussein AK. Experimental studies of flow boiling heat transfer by using nanofluids. Journal of Thermal Analysis and Calorimetry. 2019; 138 :125  46.
Zhang C, Zheng L, Zhang X, Chen G. MHD flow and radiation heat transfer of nanofluids in porous media with variable surface heat flux and chemical reaction. Applied Mathematical Modelling. 2015; 39 (1):165181  47.
Fang X, Chen Y, Zhang H, Chen W, Dong A, Wang R. Heat transfer and critical heat flux of nanofluid boiling: A comprehensive review. Renewable and Sustainable Energy Reviews. 2016; 62 :924940  48.
Kamel MS, Lezsovits F, Hussein AM, Mahian O, Wongwises S. Latest developments in boiling critical heat flux using nanofluids: A concise review. International Communications in Heat and Mass Transfer. 2018; 98 :5966  49.
Cheng L, Xia G, Li Q, Thome JR. Fundamental issues, technology development, and challenges of boiling heat transfer, critical heat flux, and twophase flow phenomena with nanofluids. Heat Transfer Engineering. 2018; 3 :136  50.
Moreira TA, Moreira DC, Ribatski G. Nanofluids for heat transfer applications: A review. Journal of the Brazilian Society of Mechanical Sciences and Engineering. 2018; 40 (6):303  51.
Pethkool S, EiamsaArd S, Kwankaomeng S, Promvonge P. Turbulent heat transfer enhancement in a heat exchanger using helically corrugated tube. International Communications in Heat and Mass Transfer. 2011; 38 (3):340347