Published on *PNG 520: Phase Behavior of Natural Gas and Condensate Fluids* (https://www.e-education.psu.edu/png520)

Module Goal: To introduce you to quantification in fluid phase behavior.

Module Objective: To introduce you to the concept of Z-factor and the van der Waals equation of state.

From the last module, it is very likely that one question is left in our minds. How can we adjust the ideal model to make it suitable for real gases? Well, we already have the answer. We said that once we have established a *base (ideal) model*, we look at a real case by estimating how close (or far) it performs with respect to the *base (ideal) case,* and introducing the corresponding corrections. Again, such corrections will take into account all the considerations that our original assumptions left out.

For the case of gas behavior, we introduce a correction factor to account for the discrepancies between experimental observations and predictions from our *ideal* model. This correction factor is usually referred to as the *compressibility factor* (Z), and is defined as:

$$Z=\frac{V}{{V}_{Ideal}}$$ (7.1)

In the previous equation, *V* is the real volume occupied by the gas and *VIdeal* is the volume that the *ideal* model predicts for the same conditions. The ideal volume is given by:

$${V}_{Ideal}=\frac{nRT}{P}$$ (7.2)

Hence, the equation of state for real gases is written as:

$$PV=ZnRT$$ (7.3)

Engineers are very much familiar with this equation, to the extent that it is usually recognized as *Engineering EOS*. Please note that for Z = 1, this equation collapses to the* ideal gas model*. In fact, unity is the compressibility factor of any gas that behaves ideally. However, please note that Z = 1 is a *consequence* of ideal behavior, but this is* not* a definition.

Is it possible to have a real gas at a condition at which Z=1 without being ideal (far removed from the ideal-gas theory assumptions)?

For natural gases, the most enduring method of estimating Z has been the Katz-Standing Method. However, we are now living in a computer-driven era, where thermodynamic estimations are very rarely taken from graphs or plots, as was common in the past.

Assuming an equilibrium state, the three properties needed to completely define the state of a system are pressure (P), volume (V), and temperature (T). Hence, we should be able to formulate an equation relating these 3 variables, of the form f(P,T,V)=0.

An equation of state (EOS) is a functional relationship between state variables — usually a complete set of such variables. Most EOS are written to express functional relationships between P, T and V. It is also true that most EOS are still empirical or semi-empirical. Hence, the definition:

An Equation of State (EOS) is a semi-empirical functional relationship between pressure, volume and temperature of a pure substance. We can also apply an EOS to a mixture by invoking appropriate mixing rules.

There have been a number of attempts to derive a theoretically sound EOS; but, generally speaking, not much success has been achieved along that line. As a result, we use what are known as semi-empirical EOS. Most equations of state used today are semi-empirical in nature, this being so because they are fitted to data that are available. Additionally, equations of state are generally developed for pure substances. Their application to mixtures requires an additional variable (composition) and hence an appropriate mixing rule.

The functional form of an EOS can be expressed as:

$$f(P,V,T,{a}_{k}=1,{n}_{p})=0$$ (7.4)

where a_{k} = EOS parameters.

As we stated earlier, most applicable EOS today are semi-empirical, in the sense that they have some theoretical basis but their parameters (a_{k}) must be adjusted. The number of parameters (n_{p}) determines the category/complexity of the EOS. For instance, 1-parameter EOS are those for which n_{p} = 1, and 2-parameter EOS are those for which n_{p} = 2. The higher “n_{p}” is, the more complex is the EOS. Also, in general terms, the more complex the EOS, the more accurate it is. However, this is not always the case; in some cases a rather simple EOS can do a very good job.

Since the time of the ideal gas law (ideal gas EOS), a great number of equations of state have been proposed to describe real gas behavior. However, many of those have not passed the test of time. Only few have persisted through the years, this because of their relative simplicity. In the petroleum business, the most common modern EOS are the Peng Robinson EOS (PR EOS) and Soave-Redlich-Kwong EOS (SRK EOS). Both of these are cubic EOS and hence derivations of the van der Waals EOS, which we will be discussing next. There are other more complex EOS, although they have not yet found widespread application in our field:

- Lee Kesler EOS (LK EOS)
- Benedict-Webb-Rubin EOS (BWR EOS)
- Benedict-Webb-Rubin-Starling EOS (BWRS EOS)

In the natural gas business, especially in the gas transmission industry, the standard EOS used is the AGA EOS; this is an ultra-accurate EOS for Z-factor calculations — a very sensitive variable for custody-transfer operations.

We want to use EOS as the basis for generating data: volumetric data, thermophysical data, and to help us perform vapor/liquid equilibrium (VLE) calculations. Probably there has not been an area of thermodynamics to which so many hours of research have been devoted as that of the topic of EOS. Among the properties derived from an EOS include:

- Densities (vapor and liquid),
- Vapor pressures of pure components,
- Critical pressures and temperatures for the mixture,
- Vapor-Liquid equilibrium (VLE) information,
- Thermodynamic properties $(\Delta H,\Delta S,\Delta G,\Delta A)$ .

**PERIOD 1: “Foundational work”**

*Before 1662, there was an incomplete understanding and qualitative representation of the volumetric behavior of gases.*

1662: First breakthrough — Boyle’s Law. Boyle did not define an ideal behavior. When he proposed this law, he was convinced that it applied to all gases. Among the limitations that Boyle was working under: no high pressure measurements could have been taken using the equipment of his time, and his working fluid was air. Hence, it is no wonder that everything these pioneers did pertained to what we now recognize as the ideal state.

$$V\alpha \frac{1}{P}$$PV = constant

1787: Charles’ Law. It was one hundred plus years until a new, important development in the gas behavior field. Charles postulated that the volume of a gas is proportional to its temperature at isobaric conditions.

$$V\text{}\alpha \text{}T$$Combining, PV/T = constant = R, gas constant

1801: Dalton introduced the concept of partial pressures and recognized that the total pressure of a gas is the sum of the individual (partial) contributions of its constituents.

1802: Gay-Lussac. He helped to define the *universal* gas constant “R”. Dalton had looked at different gases and calculated the ratio PV/T to verify that it was constant. However, it was believed that each gas may have its own R. Gay-Lussac showed that a single constant applied to all gases, and calculated the “universal” gas constant.

1822: Cagniard de la Tour. He discovered the critical state (critical point) of a substance.

1834: Clapeyron. He was the first to suggest PV=R(T+273).

**PERIOD 2: “Monumental Work”**

*Period of turning points and landmarks with quantitative developments.*

1873: van der Waals. With van der Waals, a quantitative approach was taken

for the first time. He was an experimentalist and proposed the continuity of gases and liquid that won for him a Nobel Prize. He has provided the most important contribution to EOS development.

1875: Gibbs, an American mathematical physicist, made the most important contributions to the thermodynamics of equilibrium in what has been recognized as a monumental work.

1901: Onnes theoretically confirmed the critical state.

1902: Lewis defined the concept of fugacity.

1927: Ursell proposed a series solution (polynomial functional form) for EOS: P = 1 + b/V + c/V2 + d/V3 +… This is known as the virial EOS. Virial EOS has better theoretical foundation than any other. However, cubic EOS (as vdW’s) need only 2 parameters and have become more widespread in use.

**PERIOD 3: “Incremental Improvement”**

*During this last and current period, a better quantitative description of volumetric behavior has been achieved at a rather low pace. What is striking, as we will study later, is that most of the tools of most critical use for us today are based on the works of van der Waals, Gibbs, and Lewis, and have been around for years*.

1940: Benedit, Webb, & Rubbin proposed what can be called the “Cadillac” of EOS, i.e., the most sophisticated and most accurate for some systems. However, the price to pay is that it is complicated and not easy to use.

1949: Redlich & Kwong introduced a temperature dependency to the attraction parameter “a” of the vdW EOS.

1955: Pitzer introduced the idea of the “acentric factor” to quantify the non-sphericity of molecules and was able to relate it to vapor pressure data.

1972: Soave modified the RK EOS by introducing Pitzer’s acentric factor.

1976: Peng and Robinson proposed their EOS as a result of a study sponsored by the Canadian Gas Commission, in which the main goal was finding the EOS best applicable to natural gas systems.

Since then, there has not been any radical improvement to SRK and PR EOS, although a great deal of work is still underway.

Even though van der Waals EOS (vdW EOS) has been around for more than one hundred years, we still recognize van der Waals’ achievements as crucial in revolutionizing our thinking about EOS. We talk about vdW EOS because of pedagogical reasons, not because it finds any practical application in today’s world. In fact, vdW EOS is *not* used for any practical design purposes. However, most of the EOS being used widely today for practical design purposes have been derived from vdW EOS.

The contributions of vdW EOS can be summarized as follows:

- It radically improved predictive capability over ideal gas EOS,
- It was the first to predict continuity of matter between gas and liquid,
- It formulated the Principle of Corresponding States (PCS),
- It laid foundations for modern cubic EOS.

In his PhD thesis in 1873, van der Waals proposed to semi-empirically remove the main key “weaknesses” that the ideal EOS carried with it. Essentially, what he did was to look again at the basic assumptions that underlie the ideal EOS, which we have listed above.

vdW accounted for the non-zero molecular volume and non-zero force of attraction of a real substance. He realized that there is a point at which the volume occupied by the molecules cannot be neglected. One of the first things vdW recognized is that molecules must have a finite volume, and that volume must be subtracted from the volume of the container. At the same time, he modified the pressure term to acknowledge the fact that molecules do interact with each other though cohesive forces. These are the two main valuable recognitions that he introduced.

The ideal EOS states:

$$(P)(v)=nRT$$ (7.5)

or,

$$(P)(\stackrel{-}{v})=RT$$ (7.6)

where $\stackrel{-}{v}$ of the substance.

vdW focused his attention on modifying the terms “P” and “v” in the original ideal gas law by introducing an appropriate correction. Looking back at the inequality of equation (6.4) in Module 6, vdW proposed that the difference of both pressures is the result of the attraction forces — cohesive forces — neglected in the ideal model (equation 7.6) and thus,

$${P}_{ideal}={P}_{real}+\delta {P}_{attraction}$$ (7.7)

At this point, vdW postulated the term $\delta $Pattraction, an inverse function of the mean distance between molecules — this being a direct consequence of Newton’s law of inertial attraction forces, F $\alpha $ (distance)^{-2}. Recognizing that the volume of the gas is a measure of the mean distance between molecules (the smaller the volume, the closer the molecules and vice versa),

(7.8)

and using “a” as a constant of proportionality,

$${P}_{ideal}={P}_{real}+\frac{a}{\stackrel{-}{{v}^{2}}}$$ (7.9)

Next, vdW took care of the inequality in equation (6.5) (see Module 6). Any particle occupies a physical space; hence, the space available to the gas is less than the total volume of its container. Let us say we can experimentally determine the actual physical space that all the molecules in the container occupy, and that we call it “b”, or the co-volume. vdW then proposed:

$${\stackrel{-}{v}}_{available}={\stackrel{-}{v}}_{total}-b$$ (7.10)

The inclusion of a parameter “b” (co-volume) recognizes the role of repulsive forces. Repulsive forces prevent molecules from “destroying” one another by not letting them get too close. In a condensed state, there is a maximum allowable “closeness” among molecules. Therefore, it is because of repulsive forces that we cannot compress the volume of a fluid beyond its co-volume value “b”. If the molecules get too close to each other, repulsion forces take over to prevent their self-destruction.

In summary, vdW proposed to correct the pressure and volume terms of the ideal model represented by equation (7.6). The “new” modified-ideal equation of state becomes:

$$\left(P=\frac{a}{\stackrel{-}{{v}^{2}}}\right)(\stackrel{-}{v}-b)=RT$$ (7.11a)

or,

$$P=\frac{RT}{\stackrel{-}{v-b}}-\frac{a}{\stackrel{-}{{v}^{2}}}$$ (7.11b)

where:

P = absolute pressure

v = molar volume

T = absolute temperature

R = universal gas constant

Equation (7.11b) demonstrates that vdW EOS is explicit in pressure. At this stage it is important to stress that any prediction for $\stackrel{-}{v}<b$ from equations (7.11) is meaningless due to the physical significance that we have attached to this parameter. Since all the others parameters are constants, equations (7.11) are functional relationships of the variables P, T and $\stackrel{-}{v}$:

$$f(P,\stackrel{-}{v,}T)=0$$ (7.12)

Therefore, equation (7.11a) expresses a PVT relationship and hence, it is an equation of state. In this equation, “a” and “b” are constants that are specific to each component. However, the numerical value of “R” depends on the system of units chosen, as we discussed above.

It is time to ask ourselves a very important question:

*How do we calculate “a” and “b” for each substance? *

As can be inferred from their definitions, “a” and “b” are different for different substances; i.e.,

$${a}_{C{H}_{4}}\ne {a}_{{C}_{2}{H}_{6}}\ne {a}_{{C}_{3}{H}_{8}};{b}_{C{H}_{4}}\ne {b}_{{C}_{2}{H}_{6}}\ne {b}_{{C}_{3}{H}_{8}}$$

How do we relate “a” and “b” to well-known and easily-obtainable physical properties of substances?

It turns out that there are a set of conditions called *criticality conditions *that must be satisfied for all systems, provided that those systems satisfy the 2^{nd} law of thermodynamics. Indeed, in the previous chapter we recognized that the critical isotherm of a pure substance has a point of inflexion (change of curvature) at the critical point. Furthermore, we recognized the critical point to be the maximum point (apex) of the P-V envelope. This condition of horizontal inflexion of the critical isotherm at the critical point is mathematically imposed by the expression:

$${\left(\frac{\partial P}{\partial \stackrel{-}{v}}\right)}_{{p}_{c\text{'}}{T}_{c}}={\left(\frac{{\partial}^{2}P}{\partial {\stackrel{\sim}{v}}^{2}}\right)}_{{p}_{c\text{'}}{T}_{c}}=0$$ (7.13)

These conditions are called the *criticality conditions.* It turns out that when one imposes these conditions on equation (7.11a), one is able to derive expressions for the parameters “a” and “b” as a function of critical properties as follows:

$$a=\frac{27}{64}\frac{{R}^{2}{T}_{c}^{2}}{{P}_{c}}$$ (7.14a)

$$b=\frac{R{T}_{c}}{8{P}_{c}}$$ (7.14b)

You may want to prove this as an exercise. “a” and “b” can therefore be known because they are functions of known (tabulated) properties of all substances of interest (critical pressure and temperature.)

So far, we have applied vdW EOS to pure components. Can we extrapolate this to apply to multi-component systems?

To extend this concept to a system of more than one component, we use what is called a *mixing rule.* A mixing rule relates the parameters that characterize the mixture (a_{m} and b_{m}) to the individual contributions of the pure components that make up that mixture (a_{i} and b_{i})

How do we do this? vdW proposed to weight the contributions of each component using their mole compositions, as follows:

$${a}_{m}=\sum _{i}\sum _{j}{y}_{i}{y}_{j}\sqrt{{a}_{i}}{a}_{j}$$ (7.15a)

$${b}_{m}=\sum _{i}{y}_{i}{b}_{i}$$ (7.15b)

The former is called the quadratic mixing rule, while the latter is known as the linear mixing rule.

Answer the following problems, and submit your answers to the drop box in Canvas that has been created for this module.

*Please note:*

- Your answers must be submitted in the form of a Microsoft Word document.
- Include your Penn State Access Account user ID in the name of your file (for example, "module2_abc123.doc").
- The due date for this assignment will be sent to the class by e-mail in Canvas.
- Your grade for the assignment will appear in the drop box approximately one week after the due date.
- You can access the drop box for this module in Canvas by clicking on the Lessons tab, and then locating the drop box on the list that appears.

- If a gas
*is*ideal, would its compressibility factor (Z) be always equal to one? - For a gas with Z=1, would its behavior be ideal?
- Take a look at the Standing-Katz Compressibility Factor Plot for Natural Gases. What is the information you need to obtain a value of “Z” for a gas? What happens to “Z” at low pressures? What is the behavior of “Z” at high pressures? What is the compressibility factor of Methane (P
_{c}= 666 psia, T_{c}= – 117 F) at P = 1000 psia and T = 0 F? - As pressure approaches to zero, what should vdW EOS collapse to? Why? Can you show it?
- Speculate on how you could calculate the “Z” factor for a gas using vdW EOS.