Chapter 2. Literature Review
2.5 Overview of Continuous Thermodynamics
The representation of the composition of any multicomponent fuel is always difficult in engineering applications such as distillation and other separation techniques. Some examples of commonly used multicomponent mixtures in engineering are; petroleum fractions, coal derived chemicals, acids and aldehydes in biomass oil. The representation of two or three components is easy and can be represented by mole fractions, but when they contain more than hundreds or thousands of different components, their representation is extremely difficult. In addition, mole fractions of most of the components in multicomponent mixtures are unknown. Hence, there should be some technique that can be applied to accurately represent these fuels. In the literature, three techniques are found, namely; (1) pseudo component technique, (2) continuous thermodynamics and (3) semi continuous thermodynamics. Sometimes the continuous thermodynamics technique is also called thermodynamics of continuous mixture.
In the pseudo component technique, individual components with similar properties are lumped together to form one group, then their representation is carried out by a block or bar type distribution (see Figure 2-9). This technique is useful for representing the binary mixture or mixture with small component numbers but it is not adequate for representing multicomponent fuels. Generally, this technique is adopted to avoid mathematical complexity in phase equilibrium calculations, but it has a disadvantage of distortion in expressing composition accurately [90].
The second technique (continuous thermodynamics technique) uses a simple continuous mathematical distribution function (probability density function (PDF)) to represent one of the physical properties of the fuel instead of the mole fractions of individual components, as shown in Figure 2-9. The third technique is can be summarised as summation of the first two techniques. In other words, a few components of fuel are represented as discrete components and others are represented by probability density functions. This technique is very useful when some the components of the mixture are discrete (such as nitrogen in nitrogen-fuel vapour mixture) which are represented as the discrete components, while the other continuous components (fuel vapour) are represented by PDFs.
Generally, fuel is composed of many homologous chemical components and each these components are comprised of many species. For example, heavy fuel oil is comprised of n-paraffins, aromatics and naphthenes etc, yet in that mixture n-paraffins itself is comprised of many different spices such as octane, nonane, decane etc.
In continuous thermodynamics, the distribution function represents some physical properties of homologous chemical components of multicomponent mixture, such as boiling point of component, molecular weight of the component or carbon number of the component. These homologous components can be paraffins, naphthenes, aromatics etc. Figure 2-9 shows the comparison between conventional discrete (classical thermodynamics) and continuous thermodynamics representation of multicomponent fuel mixtures. As shown in Figure 2-9, in discrete modelling mole fractions of the species of homologous components are used to represent the composition of that particular component, while in continuous thermodynamic modelling these mole fractions are simply replaced by an appropriate distribution function. And depending upon the number of homologous hydrocarbons groups, the number of distributions can be used.
Figure 2-9: Discrete and continuous representation of multicomponent mixture
Conti. Thermo.
In discrete modelling summation of all individual components mole fractions are given by unity as; 1 1. H i i x = =
∑
(16)While in continuous thermodynamics the above equation for individual phase is given as [17]; 1 ( ) 1. ( ) 1. J j j j j I I x f I dI where f I dI = = =
∑ ∫
∫
(17)In continuous thermodynamics, each individual component in the fuel mixture is characterised by the characterising variable I . As mentioned earlier this characterising variable can be any physical property. Therefore, the total molar concentration of fuel mixture with distribution function is given as [91];
2 1 1 2 ( , ) ( ) I I n I I =
∫
f I dI (18)Where, n is the molar concentration of the propertyIand is between I1 andI2.
Representation of the semi continuous thermodynamics for Hfamily of discrete species and J family of continuous distribution is given as the summation of the above two approaches [17]; 1 1 ( ) 1. ( ) 1. H J i j j j i j I I x x f I dI where f I dI = = + = =
∑
∑ ∫
∫
(19)The concept of continuous thermodynamics was first introduced by Katz & Brown [92] approximately 70 years ago. They introduced an adapted method of calculating vapour
pressure of petroleum fractions. They assumed that properties of a continuous mixture vary continuously in the composition range. Subsequently, this technique was not often used for some time, but over the past two decades many researchers have used it. The majority of work using this technique has been found is in the petroleum industry, where mixtures contain large numbers of different and dissimilar components.
Some of the major areas of science and engineering found in literature, where the continuous thermodynamics technique has been successfully applied are;
• Gas condensate mixtures simulation [93] • Natural gas dew points calculations [94, 95]
• Petroleum mixture phase equilibrium [18, 90, 96-99] • Petroleum mixtures flash calculation [85, 86, 90, 100-102] • Distillation and absorption of petroleum mixtures [103-105] • Liquid –liquid phase equilibrium of polymers [85]
• Homogeneous nucleation of multicomponent vapour [106]
From the above literature three main points are found to be important for continuous thermodynamics modelling [18];
1. The choice of proper distribution function to represent the mixture 2. The choice of proper characterising variable
3. The choice of phase equilibrium
To represent the fuel, the distribution function is one of the basic requirements of continuous thermodynamics. It can be any suitable mathematical distribution function that represents the mixture accurately, but its choice depends on convenient handling throughout the model execution. In many papers [103, 107-109] a Gaussian distribution has been used as the distribution function to represent the mixture. The only limitation of this distribution is that it is unbound at both ends. Therefore, it cannot be used in practical calculations where the characterising variable requires a lower bound. Some common examples of such characterising variable are molecular weight of petroleum fractions, carbon number etc. This limitation of the Gaussian distribution can be overcome by
selecting the gamma distribution function. The gamma distribution function is simple and it has a lower bound therefore it more accurately represents petroleum fractions. Many researchers such as Cotterrman & Prausnitz [85, 86, 100], Peng et al.[99], Chou & Prausnitz [104], Hallett et al. [18-21, 61, 71, 72, 110, 111], Lippert et al. [91, 112, 113], Zhu & Reitz [75, 88], Ra & Reitz [114, 115], Harstad et al. [116, 117], Rakowski et al.[17] used the gamma distribution in their modelling. Other distribution functions found in the literature are; a bivariate log-normal distribution by Teja & William [118], a
β
- distribution function by Radosz [119] and a general distribution function by Vakili- Nezhaad et al.[120].After the choice of distribution function, the choice of characterising variable is important. This characterising variable can be any physical properties of the multicomponent mixture. Some examples of such characterising variables found in the literature are;
• Molecular weight of the component, used by Cotterman & Prausnitz [85], Peng et al.[99], Hallett et al. [18-21, 61, 71, 72, 110, 111], Lippert et al. [91, 112, 113], Zhu & Reitz [75, 88], Ra & Reitz [114, 115] and Rakowski et al.[17]
• Component’s boiling point, used by Kehlen & Ratzsch [108].
• Component’s carbon number, used by Willman & Teja [94].
The third important point to be considered for continuous thermodynamics is vapour- liquid equilibrium. Vapour-liquid equilibrium is extremely important; it provides the relationship between vapour phase and liquid phase molecules at the liquid surface and when dealing with equilibrium of hundreds and thousands of different molecules its importance is high. Two main approaches are found in the literature of continuous thermodynamics for vapour-liquid equilibrium: 1, ideal solution and 2, non-ideal solution approach. Most of the researchers, Kehlen & Ratzsch [107], Hallett et al. [18-21, 61, 71, 72, 110, 111], Lippert at al. [91, 112, 113], Zhu & Reitz [75, 88], Ra & Reitz [114, 115], Harstad at al. [116, 117], Doue at al. [121-123] used an ideal solution assumption in their respective study. In those studies, Raoult’s law was used to obtain the mole fraction (which is part of the vapour-liquid equilibrium) of different components of multicomponent mixture along with Clasious-Clapeyron equation and Trouton’s rule to
give the component’s vapour pressure. These basic assumptions yield the relationship between liquid phase and vapour phase at low pressure. However, at elevated pressure an appropriate equation of state yields the VLE. An equation of state provides the relationship between state variable (temperature, pressure, mass, enthalpy etc) of liquid and vapour phase under given physical condition. Many researchers used different types of equation of state to describe the relations between state variables which are;
• Viral equation of state [94]
• Peng-Robinson equation of state [96, 99, 124] • Van der Wales equation of state [98]
• Redlich-Kwong or Soave-Redlich-Kwong equation of state [85, 104].
Continuous thermodynamics gained popularity since a successful mathematical model for the evaporation of multicomponent fuel developed by Tamin & Hallete [18] in 1995. Thereafter a large amount of research work [17, 19, 20, 71, 72, 75, 110, 111, 114, 115, 117, 121, 123, 125] has been done on the fuel evaporation process using this technique. The continuous thermodynamics technique also has potential to describe the behaviour of a multicomponent fuel in a spray model. This was demonstrated by Lippert at al. [91, 112, 113]. Recently, Hallett & Clark [21] developed a continuous thermodynamics model of biomass evaporation and pyrolysis which shows good agreement with the experimental results.
In summary, continuous thermodynamics modelling has proved successful for dealing with multicomponent mixtures.