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The simulated result of a PFR shows that for a fix diameter of 0.05 m, by
varying the RTD between the ranges of 1 minute to 65 minutes, the conversion
of triglycerides into product is increased due to increase in length of the reactor.
At RTD = 60 minutes, the projected conversion in a PFR is 95%. In order to
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1.1 m. However, an OFBR gives greater conversion at reduced residence time
with the same diameter. It has been projected that the required length is
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reactor and the corresponding RTD
OFBR has significantly lower power density than the PFR and if compared to an
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Oscillatory flow in baffled tubes has been shown to enhance process
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ASPEN PLUS simulation also indicated that complete conversion of triglyceride
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Cambridge University Press and the Editorial Board of the Journal of Fluid Mechanics are pleased to announce a new initiative: "Focus on Fluids". Every month, one particularly interesting paper in the Journal will be the subject of an extended review and discussion by an acknowledged and invited expert in the field. The Focus on Fluids article will appear both in the print journal, and as a free online pdf from the Focus on Fluids website. The FoF article will explain the context, importance and implications of the underlying paper to a wider scientific audience, highlighting both the key findings and breakthroughs of the paper and the implications of the research for future activity. The twin aims of this initiative are to raise the visibility of fluid mechanics research and to attract more people to this beautiful, challenging and increasingly relevant subject.
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Searching for articles
The sub-menu links to the left provide two ways to search for JFM articles. Published articles links to a page giving access to an index of all published papers that can be searched by author name, title keyword or volume number. Articles In Press gives a list of papers currently in the publication process; once an article has been published, it is moved from this page to the Published articles searchable index.
The is also a searchable index of Book Reviews published since 1996.
A barotropic, compressible fluid at rest is governed by the statics equation,
where z is the height above an arbitrary datum, and g is the gravity acceleration constant (9.81 m/s2; 32.2 ft/s2). This equation describes the pressure profile of the atmosphere, for example.
For an incompressible fluid, the statics equation simplifies to,
This equation describes the pressure profile in a body of water, or in a manometer.
If the fluid is compressible but barotropic, then the density and the pressure can be integrated into the "pressure per density" function , giving the following alternate form for the compressible fluid statics equation,
Note that the equation at the top of the page can still be applied though, as it makes no assumption on the fluid's equation of state.
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An equation of state is a formula describing the interconnection between various macroscopically measurable properties of a system. This document only adresses the behavior of physical states of matter, not the conversion from one state to another.
For physical states of matter, this equation usually relates the thermodynamic variables of pressure, temperature, volume and number of atoms to one another.
In materials science the important properties are often what are termed "mechanical properties" rather than physical properties. Examples of mechanical properties would be hardness and ductility. Mechanical properties will not be addressed here.
Gas - There are several types of gases with slightly different behaviors. These are ideal gasses, real gasses, super critical fluids, plasmas and critical opalescent conditions. The ideal gas law is often used as the first order description of any gas although this practice is questionable in the case of critical opalescent conditions.
Ideal Gas - Although no gas is truly ideal, many gasses follow the ideal gas law very closely at sufficiently low pressures. The ideal gas law was originally determined empirically and is simply
p V = n R T
p = absolute pressure (not gage pressure)
V = volume
n = amount of substance (usually in moles)
R = ideal gas constant
T = absolute temperature (not F or C)
where some values for R are
8.3145 J mol-1 K-1
0.0831451 L bar K-1 mol-1
82.058 cm3 atm mol-1 K-1
0.0820578 L atom mol-1 K-1
1.98722 cal mol-1 K-1
62.364 L Torr K-1 mol-1
Real Gas - Real gas laws try to predict the true behavior of a gas better than the ideal gas law by putting in terms to describe attractions and repulsions between molecules. These laws have been determined empirically or based on a conceptual model of molecular interactions or from statistical mechanics.
A well known real gas law is the van der Waals equation
( P + a / Vm2 )( Vm - b ) = R T
P = pressure
Vm = molar volume
R = ideal gas constant
T = temperature
where a and b are either determined empirically for each individual compound or estimated from the relations.
a = 27 R2 Tc2
--------
64 Pc
b = R Tc
----
8 Pc
Tc = critical temperature
Pc = critical pressure
The first parameter, a, is dependent upon the attractive forces between molecules while the second parameter, b, is dependent upon repulsive forces.
Another two parameter real gas equation is the Redlich-Kwong equation. It is almost always more accurate than the van der Waals equation and often more accurate than some equations with more than two parameters. The Redlich-Kwong equation is
( p + a ) ( Vm - b ) = R T
------------------
Vm ( Vm + b ) T1/2
p = pressure
a = empirical constant
Vm = molar volume
R = ideal gas constant
b = empirical constant
T = temperature
where a and b are not identical to the a and b in the van der Waals equation.
Equations of state in terms of reduced variables give reasonable results without any empirically determined constants for a specific substance. However, these are not generally as accurate as equations using empirical constants. One such equation is
( Pr + 3 / Vr2 ) ( Vr - 1/3 ) = 8/3 * Tr
Pr = reduced pressure
Tr = reduced temperature
Vr = reduced volume
where reduced pressure and temperature are the unitless quantities obtained by dividing the value by the critical value. In the case of reduced volume, molar volume is divided by critical molar volume.
A two parameter equation which is no longer used much is the Berthelot equation
p = R T - a
----- ----
V - b T V2
p = pressure
a = empirical constant
V = volume
R = ideal gas constant
b = empirical constant
T = temperature
A somewhat more accurate modified Berthelot equation is
p = R T [ 1 + 9 p Tc ( 1 - 6 Tc2 ) ]
--- -------- -----
V 128 Pc T T2
p = pressure
Pc = critial pressure
V = volume
R = ideal gas constant
T = temperature
Tc = critical temperature
The Dieterici equation is another two parameter equation which has been seldom used in recent years.
p = R T e-a / ( Vm R T )
------------------
Vm - b
p = pressure
a = empirical constant
Vm = molar volume
R = ideal gas constant
b = empirical constant
T = temperature
The Clausius equation is a simple three parameter equation of state.
[ P + a ] ( Vm - b ) = R T
-------------
T ( Vm + c )2
a = Vc - R Tc
----
4 Pc
b = 3 R Tc - Vc
------
8 Pc
c = 27 R2 Tc3
--------
64 Pc
P = pressure
T = temperature
R = real gas constant
Vm = molar volume
Tc = critical temperature
Vc = critical volume
Tc = critical temperature
The virial equation is popular because the constants are readily obtained using a perturbative treatment such as from statistical mechanics. The virial coeficients are also readily fitted to experimental data because it is a linear curve fit.
p Vm = R T ( 1 + B(T) / Vm + C(T) / Vm2 + D(T) / Vm3 + ... )
or
p Vm = R T ( 1 + B'(T) / p + C'(T) / p2 + D'(T) / p3 + ... )
p = pressure
Vm = molar volume
R = ideal gas constant
T = temperature
B, C, D, .. = constants for a given temperature
B', C', D', .. = constants for a given temperature
where B is not identical to B' and etc.
The equation of state created by Peng and Robinson has been found to be useful for both liquids and real gasses.
p = R T - a ( T )
------ ----------------------------
Vm - b Vm ( Vm + b ) + b ( Vm - b )
p = pressure
a = empirical constant
Vm = molar volume
R = ideal gas constant
b = empirical constant
T = temperature
The Wohl equation is formulated in terms of critial values making it a bit more convenient for situations where no real gas constants are available
[ p + a - c ] ( Vm - b ) = R T
--------------- ------
T Vm ( Vm - b ) T2 Vm3
a = 6 Pc Tc Vc2
b = Vc / 4
c = 4 Pc Tc2 Vc3
p = pressure
Vm = molar volume
R = ideal gas constant
T = temperature
Pc = critical pressure
Tc = critical temperature
Vc = critical volume
A some what more complex equation is the Beattie-Bridgeman equation
P = R T d + ( B R T - A - R c / T2 ) d2 + ( - B b R T + A a - R B c / T2 ) d3
+ R B b c d4 / T2
P = pressure
R = ideal gas constant
T = temperature
d = molal density
a, b, c, A, B = empirical parameters
Benedict, Webb and Rubin suggest the real gas equation of state
P = R T d + d2 { R T [ B + b d ] - [ A + a d - a alpha d4 ]
- 1 [ C - c d ( 1 + gama d2 ) exp ( - gama d2 ) ] }
--
T2
P = pressure
R = ideal gas constant
T = temperature
d = molal density
a, b, c, A, B, C, alpha, gama = empirical parameters
Supercritical Fluids - Supercritical fluids are well described by real and ideal gas laws.
Critical Opalescence - Critical behavior is generally described using real gas equations which have constants defined in a way which ensures that the slope of reduced pressure vs. reduced volume is zero at the critical point. These give reasonable estimates of the relationships between pressure, volume and temperature but do not describe the opalescence or unique chemical properties very near the critical point.
Plasma - The physical behavior of plasmas is most often described by the ideal gas law equation which is quite reasonable except at very high pressures.
Liquid - Liquids are much less compressible than gasses. Even when a liquid is described with an equation similar to a gas equation, the constants in the equation will result in much less dramatic changes in volume with a change in temperature. Like wise at constant volume, a temperature change will give a much larger pressure change than seen in a gas.
A common equation of state for both liquids and solids is
Vm = C1 + C2 T + C3 T2 - C4 p - C5 p T
Vm = molar volume
T = temperature
p = pressure
C1, C2, C3, C4, C5 = empirical constants
where the empirical constants are all positive and specific to each substance.
For constant pressure processes, this equation is often shortened to
Vm = Vmo ( 1 + A T + B T2 )
Vm = molar volume
Vmo = molar volume at 0 degrees C
T = temperature
A, B = empirical constants
where A and B are positive.
The equation of state created by Peng and Robinson has been found to be useful for both liquids and real gasses.
p = R T - a ( T )
------ ----------------------------
Vm - b Vm ( Vm + b ) + b ( Vm - b )
p = pressure
a = empirical constant
Vm = molar volume
R = ideal gas constant
b = empirical constant
T = temperature
Superfluid - Superfluids are physically liquids although they have interesting properties, which are quantum mechanical in origin. Since this is still an active area of research and not completely understood, a reference to an introductory article is given, but no equations will be presented here.
Suspension - Suspensions behave physically most like liquids.
Colloid - A colloid being a type of suspension is also physically most like a liquid.
Liquid Crystal - Depending upon the temperature, liquid crystals may be crystalline, glassy, flexible thermoplastics or ordered liquids. At sufficiently high temperatures, a true liquid phase will exist. Most of the physical properties of these are the same as non liquid crystal compounds. One exception is that as liquid crystal compounds are added to a solvent the viscosity increases as expected until the concetration becomes high enough to form a liquid crystal phase, when the viscosity drops.
Visceoelastic - Since visceoelastics behave like solids on short time scales and like liquids over a long period of time, equations for liquids and solids could be used. Most of the usefulness of visceoelastics is based on their mechanical properties rather than their physical properties.
Solid - The volume of a solid will generally change very little with a change in temperature. However, most solids are very incompressible so a constant volume heating will give a very large pressure change for even a small change in temperature. Crystals, glasses and elastomers are all types of solids.
A common equation of state for both liquids and solids is
Vm = C1 + C2 T + C3 T2 - C4 p - C5 p T
Vm = molar volume
T = temperature
p = pressure
C1, C2, C3, C4, C5 = empirical constants
where the empirical constants are all positive and specific to each substance.
For constant pressure processes, this equation is often shortened to
Vm = Vmo ( 1 + A T + B T2 )
Vm = molar volume
Vmo = molar volume at 0 degrees C
T = temperature
A, B = empirical constants
where A and B are positive.
Crystal - Crystals are solids which are often very hard. The equations above are used for describing the physical properties of crystals.
Glass - Glasses are generally very brittle. The equations above are useful for describing the physical behavior until the stress becomes too great and the material shatters.
Elastomer - An elastomer is an amorphous solid which can be deformed with out breaking. The change in volume is generally negligible with deformation. However, the cross sectional area may change considerably. For changes in temperature and pressure, elastomers can be considered to be solids although much softer than other solids.
Superplastic - The unique ability of superplastics to stretch is a mechanical property. Physically, superplastics are treated as solids.
Bose-Einstein Condensate - At the time of this writing, the first reports of having made a Bose-Einstein condensate have just been released. No measurements of physical properties have yet been made. Considering various aspects of the theory predicting the existence of this state lead to the conclusions that it might be a solid or a very supercooled gas or one very large single atom.
Refractory - Refractory materials behave physically as solids.
Further Information
For an introductory chemistry text see
L. Pauling "General Chemistry" Dover (1970)
A physical chemistry text for non-chemists is
P. W. Atkins "The Elements of Physical Chemistry" Oxford University Press (1993)
A physical chemistry text for undergraduate chemistry majors is
I. N. Levine "Physical Chemistry" McGraw-Hill (1995)
A review of real gas equations is
K. K. Shah, G. Thodos Industrial and Engineering Chemistry, vol 57, no 3, p. 30 (1965)
An introductory article about superfluids is
O. V. Lounasmaa, G. Pickett Scientific American, p. 104, June (1990)
A mathematical treatment can be found in
D. L. Goodstein "States of Matter" Dover (1985)
Properties of high molecular weight solids (most commonly polymers) are discussed in
H. R. Allcock, F. W. Lampe "Contemporary Polymer Chemistry" Prentice-Hall (1990)
Solid state properties are covered in
A. R. West "Solid State Chemistry and its Applications" John Wiley & Sons (1992)
A review article is
M. Ross, D. A. Young, Ann. Rev. Phys. Chem. 44, 61 (1993).
Return to table of contents.
Boyle's law (1662)
Boyle's Law was perhaps the first expression of an equation of state. In 1662, the noted Irish physicist and chemist Robert Boyle performed a series of experiments employing a J-shaped glass tube, which was sealed on one end. Mercury was added to the tube, trapping a fixed quantity of air in the short, sealed end of the tube. Then the volume of gas was carefully measured as additional mercury was added to the tube. The pressure of the gas could be determined by the difference between the mercury level in the short end of the tube and that in the long, open end. Through these experiments, Boyle noted that the gas volume varied inversely with the pressure. In mathematical form, this can be stated as:
The above relationship has also been attributed to Edme Mariotte and is sometimes referred to as Mariotte's law. However, Mariotte's work was not published until 1676.
[edit]Charles's law or Law of Charles and Gay-Lussac (1787)
In 1787 the French physicist Jacques Charles found that oxygen, nitrogen, hydrogen, carbon dioxide, and air expand to the same extent over the same 80 kelvin interval. Later, in 1802, Joseph Louis Gay-Lussac published results of similar experiments, indicating a linear relationship between volume and temperature:
[edit]Dalton's law of partial pressures (1801)
Dalton's Law of partial pressure states that the pressure of a mixture of gases is equal to the sum of the pressures of all of the constituent gases alone.
Mathematically, this can be represented for n species as:
[edit]The ideal gas law (1834)
In 1834 Émile Clapeyron combined Boyle's Law and Charles' law into the first statement of the ideal gas law. Initially the law was formulated as pVm=R(TC+267) (with temperature expressed in degrees Celsius), where R is the gas constant. However, later work revealed that the number should actually be closer to 273.2, and then the Celsius scale was defined with 0 °C = 273.15 K, giving:
[edit]Van der Waals equation of state
In 1873, J. D. van der Waals introduced the first equation of state derived by the assumption of a finite volume occupied by the constituent molecules.[2] His new formula revolutionized the study of equations of state, and was most famously continued via the Redlich–Kwong equation of state and the Soave modification of Redlich-Kwong.
[edit]Major equations of state
For a given amount of substance contained in a system, the temperature, volume, and pressure are not independent quantities; they are connected by a relationship of the general form:
In the following equations the variables are defined as follows. Any consistent set of units may be used, although SI units are preferred. Absolute temperature refers to use of the Kelvin (K) or Rankine (°R) temperature scales, with zero being absolute zero.
= pressure (absolute)
= volume
= number of moles of a substance
= = molar volume, the volume of 1 mole of gas or liquid
= absolute temperature
= ideal gas constant (8.314472 J/(mol·K))
= pressure at the critical point
= molar volume at the critical point
= absolute temperature at the critical point
[edit]Classical ideal gas law
The classical ideal gas law may be written:
The ideal gas law may also be expressed as follows
where ρ is the density, γ = Cp / Cv is the adiabatic index (ratio of specific heats), e = CvT is the internal energy per unit mass (the "specific internal energy"), Cv is the specific heat at constant volume, and Cp is the specific heat at constant pressure.
[edit]Cubic equations of state
[edit]Van der Waals equation of state
The Van der Waals equation of state may be written:
where Vm is molar volume, and a and b are substance-specific constants. They can be calculated from the critical properties pc,Tc and Vc (noting that Vc is a the molar volume at the critical point) as:
Also written as
Proposed in 1873, the van der Waals equation of state was one of the first to perform markedly better than the ideal gas law. In this landmark equation a is called the attraction parameter and b the repulsion parameter or the effective molecular volume. While the equation is definitely superior to the ideal gas law and does predict the formation of a liquid phase, the agreement with experimental data is limited for conditions where the liquid forms. While the van der Waals equation is commonly referenced in text-books and papers for historical reasons, it is now obsolete. Other modern equations of only slightly greater complexity are much more accurate.
The van der Waals equation may be considered as the ideal gas law, “improved” due to two independent reasons:
Molecules are thought as particles with volume, not material points. Thus V cannot be too little, less than some constant. So we get (V − b) instead of V.
While ideal gas molecules do not interact, we consider molecules attracting others within a distance of several molecules' radii. It makes no effect inside the material, but surface molecules are attracted into the material from the surface. We see this as diminishing of pressure on the outer shell (which is used in the ideal gas law), so we write (p + something) instead of p. To evaluate this ‘something’, let's examine an additional force acting on an element of gas surface. While the force acting on each surface molecule is ~ρ, the force acting on the whole element is ~ρ2~.
[edit]Redlich–Kwong equation of state
Introduced in 1949 the Redlich–Kwong equation of state was a considerable improvement over other equations of the time. It is still of interest primarily due to its relatively simple form. While superior to the van der Waals equation of state, it performs poorly with respect to the liquid phase and thus cannot be used for accurately calculating vapor-liquid equilibria. However, it can be used in conjunction with separate liquid-phase correlations for this purpose.
The Redlich–Kwong equation is adequate for calculation of gas phase properties when the ratio of the pressure to the critical pressure (reduced pressure) is less than about one-half of the ratio of the temperature to the critical temperature (reduced temperature):
[edit]Soave modification of Redlich-Kwong
Where ω is the acentric factor for the species.
for hydrogen:
In 1972 Soave replaced the 1/√(T) term of the Redlich-Kwong equation with a function α(T,ω) involving the temperature and the acentric factor. The α function was devised to fit the vapor pressure data of hydrocarbons and the equation does fairly well for these materials.
Note especially that this replacement changes the definition of a slightly, as the Tc is now to the second power.
[edit]Peng-Robinson equation of state
In polynomial form:
where, ω is the acentric factor of the species, R is the universal gas constant and Z=PV/(RT) is compressibility factor.
The Peng-Robinson equation was developed in 1976 in order to satisfy the following goals:[3]
The parameters should be expressible in terms of the critical properties and the acentric factor.
The model should provide reasonable accuracy near the critical point, particularly for calculations of the compressibility factor and liquid density.
The mixing rules should not employ more than a single binary interaction parameter, which should be independent of temperature pressure and composition.
The equation should be applicable to all calculations of all fluid properties in natural gas processes.
For the most part the Peng-Robinson equation exhibits performance similar to the Soave equation, although it is generally superior in predicting the liquid densities of many materials, especially nonpolar ones. The departure functions of the Peng-Robinson equation are given on a separate article.
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