In the Classroom
Theoretical Insights for Practical Handling of Pressurized Fluids Alfonso Aranda* and María del Prado Rodríguez Facultad de Químicas, Paseo de la Universidad s/n, Ciudad Real, 13071, Spain; *
[email protected] Students generally recognize the value of applying chemistry in a real-world setting. In this article we try to reinforce, rather than introduce, chemical topics such as liquid–vapor equilibrium, equation of state, and critical conditions through application to the handling of fluids. Many of the practical scenarios discussed in a chemistry or chemical engineering course use solid or liquid reactants. Problems may arise if the application involves compressed or liquefied gases. We introduce this subject to the students from two different viewpoints: the user and the supplier of pressurized fluids. As an inexperienced user, for a given application in the laboratory requiring pressurized fluids, the student may be not sure of which kind of cylinder to use, about the size, the pressure inside, the equipment, the physical state of the chemicals, and so forth. Although information about each chemical and various fluid-handling equipment may be found in the printed or Web catalogues of the supplier (1, 2), students need some insights to understand the information. From the point of view of the supplier, the storage and transport of chemical substances must be considered. For example, for economic reasons, it is necessary to reduce the volume of the container used to store and transport the compressed fluids, which may result in problematic levels of pressure. This article tries to provide the key ideas to be considered when handling pressurized fluids. We present three typical examples that may be used in lectures. Then, the instructor may propose additional problems to be solved as a user or supplier including problems concerning the design of the cylinders for fluids. Some examples are suggested at the end of the article. Our experience shows that although such “design problems” generally require a greater effort, they are appro-
liquid
C critical point
P
B isochore solid
orthobaric
priate for small-group work and result more in-depth learning. Furthermore, the students develop secondary skills, such as the selective search of data, identification of freedom degrees, and the setting-up of viability criteria. From a theoretical point of view, this exercise requires little background knowledge: just several details about the phase diagram of a pure substance, as shown in Figure 1. We assume that the students recognize the different zones, solid, liquid, and gas states, and the equilibrium curves, solid–liquid, solid–vapor, and liquid–vapor, which depend on the pressure and temperature conditions (P, T ). For a given system in liquid–vapor equilibrium, if we increase the temperature maintaining both phases (climbing up the liquid–vapor curve), an increase of density in the vapor phase is observed associated with a decrease in density in the liquid phase. At a temperature specific to each substance (the critical temperature, Tc ) the densities of both phases become the same. This also occurs with the intensive properties, that is, at Tc we have only one phase and strictly we cannot say whether it is a liquid or a gas. At temperatures above Tc liquid and vapor phases cannot coexist in equilibrium and isothermal compression of the vapor will not cause condensation, in contrast to compression below Tc. Within this context, we define a fluid as being liquid if T < Tc and the molar volume V < Vc. We call vapor a fluid that is susceptible to condensation by an increase of pressure at constant T; that is, it is required that T < Tc and V > Vc. Supercritical fluids are those with T > Tc and P > Pc. For further details on phase diagrams see references 3–6. Guidelines for the Use of Pressurized Fluids
Temperature When storing chemicals, most applications are designed to operate at room temperature since it is cheaper and does not require additional equipment for temperature control (unless there is any problem with the stability of the chemicals). Constant Volume In most of cases, chemicals are stored in rigid containers of constant volume (In fact, the container has a very low variation of volume owing to temperature fluctuations). Under such premise, an important point to be considered is the variation of the fluid volume with P and T:
A
β =
vapor
TC
T Figure 1. Phase diagram of a pure substance.
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1 V
κ= −
1 V
∂ V ∂ T
= P
∂ V ∂ P
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= − T
∂ lnV ∂ T
volumetric expansivity
P
∂ lnV ∂ P
T
isothermal compressibility
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From these partial derivatives it is easily found dV = β dT − κ dP V
and at constant V ∂ P ∂ T
= V
β κ
For an ideal gas, one can easily derive β =
1 1 ;κ = ; T P
∂ P ∂ T
= V
R P = V T
For a real gas, (∂P兾∂T )V may be obtained from an accurate equation of state but for a quick estimate we will assume a similar result as for an ideal gas. If P is not very high nor T very low, the molar volume of gases, V, is generally of large magnitude, so (∂P兾∂T )V is generally low. For example, for an ideal gas at 20 ⬚C and 1 bar, ∂ P ∂ T
= 3.4 × 10 −3 V
bar K
that is, under such conditions at constant volume, an increase of T of 1 degree gives a very low increase of P (3.4 × 10᎑3 bar). For many liquids the experimental values of β and κ are available in the literature. For example, for acetone at 20 ⬚C and 1 bar β = 1.46 × 10᎑3 K᎑1 and κ = 1.26 × 10᎑4 bar᎑1 (7), which gives ∂ P ∂ T
= 11.6 V
bar K
Therefore, under such conditions at constant volume, an increase of 1 K gives a very high increase of P (11.6 bar) and the risk of explosion arises. This behavior is given by the isochore path (A–B) in Figure 1. For such reasons [rigid containers and (∂P兾∂T )V values], liquids are not stored or transported in completely filled containers. Diurnal or seasonal temperature fluctuations of the order of 10–30 K would mean increases of P in the order of several hundred bars. So, if we must manage a liquid, we generally leave a free space for the gas resulting from the evaporation of the liquid. Under such conditions, the volume of the liquid phase may change slightly owing to temperature fluctuations at the expense of the vapor phase without dangerous increases of pressure (orthobaric path, A–C, in Figure 1), maintaining constant the total volume of the fluid inside the container. Thus, we store liquids like acetone, ethanol, water, and so forth in containers always leaving free space for the vapor phase (usually 5–30% of the total volume).
Total Volume and Pressure Suppose we must supply a gas for a specific application. If we send it under standard conditions we may require a whole cistern to carry just a few grams or kilograms since gas phase is not dense. To avoid that, gases are compressed and filled in cylinders under pressures generally up to 200 bars. In some cases, for substances that are gases at 1 bar and room temperature, when compressing at constant T, the gases 94
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condense well before 200 bar total pressure is reached. In those cases, we fill the cylinders or containers under vapor pressure conditions (not higher) at room temperature with the dense liquid phase occupying approximately 70–95% of the total volume. We then have a high amount of substance in a reduced volume container. That is useful and economic in practice. If we wanted to continue the filling process, we would have a pressure equal to vapor pressure until the whole volume is occupied by the liquid. Any further addition is unsafe as already explained; the pressure of the “liquefied gas” would sharply increase. Furthermore, that increase of P would not significantly increase the amount of substance in the cylinder since (∂V兾∂P )T is very small for liquids. Examples of such “liquefied gases” are butane, ethane, ethene, and CO2. From a theoretical point of view, the treatment of our “liquids” and our “liquefied gases” is the same: in both cases we have a pure substance in liquid–vapor equilibrium. However, the vapor pressure at room temperature is different. If it is low, that is, well below atmospheric pressure as is the case for water, we can open the cylinder and the fluid is not rapidly lost by evaporation or boiling. On the other hand, if the vapor pressure is greater than one atmosphere (butane) when we opened the cylinder the fluid would boil abruptly, creating different problems and security risks. For some fluids with vapor pressures in the range of 0.4 to 1.0 bar (still below 1 atm) the losses of the fluid by vaporization may still be considered as high. Then, these chemicals are advised to be handled with care with the aid of extractors, valves, special fittings, et cetera. Therefore these fluids may appear in the catalogues as “liquefied gas”. In the catalogues of various manufactures we may find other definitions of these terms. For example, some vendors define a compressed gas as any gas with a critical point below ᎑10 ⬚C and a liquefied gas as any gas with a critical point above ᎑10 ⬚C. Three Examples of Pressurized Fluids Let us now inspect three examples: chloroform, ethane, and nitrogen. We need data that can be found in the literature (for example, refs 7–9) such as the vapor pressure–temperature data and the critical point coordinates. Alternatively, vapor pressure data could be computed using an equation of state (9–15). The critical constants and acentric factors, ω, for the three substances are summarized in Table 1. The acentric factor is a pure-component constant commonly used for property estimations,
ω = − log
P (0.7Tc ) Pc
−1
where P(0.7Tc) is the vapor pressure at T = 0.7Tc. Table 1. Critical Constants and Acentric Factor Substance
Pc/bar
Tc/K
V c/ (cm3 mol᎑1)
ω
Chloroform
55.0
536.5
240.0
---
Ethane
48.8
305.4
145.5
0.099
Nitrogen
33.9
126.2
090.1
0.040
NOTE: Data from ref 9.
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In the Classroom
Nitrogen
Content
At room temperature we are above the Tc for nitrogen and so it cannot be present in the liquid phase. It must then be used as a compressed gas to avoid the problem of excessive total volume. Typically, filling pressures of 50, 100, 200 bars are used. The container must be able to withstand the pressure inside under the usual conditions but also with extra safety margins. So, rigid metallic cylinders are used. Since our application in the laboratory may need the gas at pressures of only a few bars or even at one atmosphere, a pressure regulator is required as special equipment. It reduces the pressure from that inside the cylinder to that required in the laboratory (giving the measurement of both pressures) and it regulates the flow of gas avoiding violent outputs. Complementary valves, fittings, tubing, and so forth are also required.
An important question is how to calculate the amount of substance we have in our container.
Nitrogen For gas phases we may use a cubic equation of state. The Peng–Robinson equation (12) is recognized to give good results for both the gas and the liquid phases (calculations for the liquid phase are, in general, with this and other equations, less accurate than the results for gases). Other equations of state, their applicability fields, and comparisons between them are reviewed in refs 9, 13–15. We use the Peng– Robinson equation in its polynomic form for the compressibility factor, Z:
( ) ( AB − B
(
)
Z 3 − 1 − B Z 2 + A − 3B 2 − 2 B Z
Ethane From the vapor pressure–temperature data in the literature (7, 8) the corresponding Clausius–Clapeyron (or Antoine) equation may be obtained
ln (Pvap / Pa) = 21.33 −
1812 T /K
where Pvap is the pressure of the vapor in units of Pa and temperature is in units of K. This equation can be used to estimate the vapor pressure at room temperature (20 ⬚C): Pvap = 37.9 bar. This fluid must then be stored in liquid–vapor equilibrium at P = 37.9 bar, leaving up to 30% for the gas phase. It is not desirable to increase the pressure because it would mean that only liquid phase is present with the corresponding risks. Since liquid phase is already present, the amount of ethane inside is high enough to be useful from a practical point of view. In these cases rigid metallic cylinders are also required owing to the high pressure. Since the pressure inside the container is higher than atmospheric pressure we must not freely open it. The recommended equipment is similar to the case of compressed gases. For other liquefied gases with vapor pressures below around 5 bars, the use of valves may replace the expensive pressure regulators if we do not require an accurate output flow.
Chloroform As in the case of ethane before, we obtain (7, 8) the Clausius–Clapeyron equation,
ln (Pvap / Pa) = 22.28 −
3616 T /K
and use it to estimate the vapor pressure at room temperature (20 ⬚C): Pvap = 0.21 bar. Since it is well below atmospheric pressure, the “liquid” will not boil at room temperature, the vaporization is not expected to be high and is not especially dangerous. Therefore, no special measures need to be taken to handle this “liquid” and we can store chloroform at room temperature. If we fill the plastic or glass bottle under free atmosphere, the total pressure will be 1 atm due to the air present with a partial pressure of 0.21 bar from the chloroform. We will leave, for example, 10% of the total volume free for the gas phase. www.JCE.DivCHED.org
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−
2
− B
3
)=
0
To simplify and generalize the cubic Peng–Robinson equation, the terms A and B are used. As shown in the appendix, A and B are functions of P, T, and the nature of the chemical. Once the values of A and B are calculated, the cubic equation in terms of Z may be solved by an iterative strategy such as the standard Newton–Raphson (16) method. In general, the values of Z fall between 0 and 1. If we are searching the value of Z in the gas phase, an initial guess of Z = 1 converges rapidly to Z. In our example Z is 1.024 for nitrogen. From this value and the relationship, Z = PV兾RT, we obtain the molar volume V = 0.125 × 10᎑3 m3 mol᎑1. Thus a standard 50-L cylinder at 200 bar and 20 ⬚C contains approximately 400 mol (11.2 kg) of nitrogen.
Ethane Consider the situation of an ethane cylinder at 20 ⬚C and 37.9 bar with a liquid–vapor equilibrium. Note that the amount of ethane in the vapor phase may not be negligible since the pressure is high. The Peng–Robinson equation also applies to such a system. Since the equation is cubic in Z, when solved for liquid–vapor equilibrium conditions three real roots are found. The smaller one ( Zliq ) is the value of the liquid phase and the greatest ( Zvap) is that of the gas phase. The intermediate root has no physical meaning. To get the vapor result we use an initial guess of Z = 1, as before. If we use an initial guess of Z = 0, the Newton–Raphson method almost always converges to the value of Z of the liquid phase. In this example the results are: Zvap = 0.516, and Vvap = 3.32 × 10᎑4 m3 mol᎑1; Zliq = 0.148 and Vliq = 0.952 × 10᎑4 m3 mol᎑1. Suppose we have a 10-L cylinder with 30% of the volume in the vapor phase: it would contain 9.0 mol of ethane in the vapor phase and 73.5 mol in the liquid phase. Chloroform For the manufacturer it is easier and more accurate to obtain an experimental value of the molar volume of a conventional liquid by directly measuring the density of the fluid in the laboratory. If we are the user, we just need to consult the catalogue for such data. Otherwise we could calculate this magnitude using the Peng–Robinson equation with depart
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value of Z = 0, like in the example of ethane. Since the vapor pressure is very low for chloroform, the number of moles in the vapor phase is negligible. Additional Considerations With a compressed gas, one always has a direct knowledge of the remaining content from the pressure measured in the gauge of the pressure regulator, and we can foresee the replacement of the cylinder. Note that this measurement is not so useful in the case of a liquefied gas. With one component and two phases we have only one degree of freedom, the temperature. Pressure is then fixed at room temperature: the vapor pressure. Therefore, the pressure inside the cylinder will be the same from the day we order it to a final situation with only a drop of liquid. We should have a second cylinder ready for a sudden need of replacement. For safety reasons, we are generally advised to store gas cylinders in vented places shielded from light. In the case of liquefied gases, especially if they have critical points only slightly above room temperature, that is strongly recommended. That is the case of ethane, Tc = 305.4 K. In a sunny, relatively hot day, room temperature may exceed Tc and a complete vaporization to the gas phase could occur with a significant increase of pressure. Although commercial cylinders should be able to resist the pressure, we must guard against unnecessary risks. In the everyday life we may find many examples were compressed and liquefied gases are used: CO2 in extinguishers, O2 to assist respiratory problems in hospitals, acetylene for soldering applications, butane and propane cylinders in the kitchen, helium for filling balloons, and so forth. This article may help to better understand their use. Using the previous material, the instructor can now propose some relatively easy calculations: • Obtain the pressure and the amount of butane inside a cylinder for domestic use. • As pair work or small-group work, consult the WWW catalogues or refs 1 and 2 for several chemicals and try to understand the reported information. • Small groups of students may work on the design of the cylinders required for the storage of ethene, argon, and bromine. Predict the physical state and estimate the pressure and the total volume for a given amount of substance. Verify that the results are consistent with the corresponding data in the catalogs of chemicals or refs 1 and 2. If the cylinders are exposed to changes of temperature from 293 to 330 K, what pressure should the containers be able to withstand in each case?
Conclusion
Journal of Chemical Education
1. Praxair Home Page. http://www.praxair.com/ (accessed Sep 2005). 2. Airliquide Home Page. http://www.airliquide.com/en/index.asp (accessed Sep 2005). 3. Rowlinson, J. S.; Swinton, F. L. Liquids and Liquid Mixtures, 3rd ed.; Butterworths: London, 1982. 4. Gramsch, S. A. J. Chem. Educ. 2000, 77, 718. 5. Glasser, L. J. Chem. Educ. 2002, 79, 874. 6. Visak, Z. P.; Rebelo, R. P. N.; Szydlowski, J. J. Chem. Educ. 2002, 79, 869. 7. Handbook of Chemistry and Physics, 78th ed.; Lide D. R., Ed.; CRC Press Inc.: Boca Raton, FL, 1997–1998. 8. NIST Chemistry WebBook. http://webbook.nist.gov/chemistry/ (accessed Sep 2005). 9. Poling, B. E.; Prausnitz, J. M.; O´Connel, J. P. The Properties of Gases and Liquids, 5th ed.; McGraw Hill: New York, 2001. 10. Noggle, J. H.; Woo, R. H. J. Chem. Educ. 1992, 69, 811. 11. Abrol, R. J. Chem. Educ. 1995, 72, 1077. 12. Peng, D. Y.; Robinson, D. B. Ind. Eng. Chem. Fundam. 1976, 15, 59. 13. Tsonopoulos, C.; Heidman, J. L. Fluid Phase Equil. 1985, 24, 1. 14. Anderko, A. Fluid Phase Equil. 1990, 61, 145. 15. Valderrama J. O. Ind. Eng. Chem. Res. 2003, 42, 1603. 16. Elliott, R.; Lira, C. Introductory Chemical Engineering Thermodynamics; Prentice Hall: Upper Saddle River, NJ, 1999.
Appendix The definitions of A and B used in the Peng–Robinson equation are: A = aP兾(R2T2) ;
B = bP兾(RT)
where a and b are constants characteristic of each gas. If we do not find their experimental values in the literature, they may be calculated from the critical data and the acentric factor, which is available in the literature for most common chemicals:
b = 0.07780 R
Tc Pc
a (T ) = a c α where a c = 0.457 R 2
Tc 2 Pc
and
We have presented key ideas and calculations to quantitatively deal with the management of chemicals under pressures above 1 atmosphere. These considerations are helpful to properly handle the technical devises and the equipment required in the use of compressed and liquefied gases.
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Literature Cited
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( ) (0.37464 + 1.5422 ω − 0.26992 ω )
α 0.5 = 1 + 1 − Tr 0.5 ×
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