Review Article  Open Access
A Reconciliation of Packed Column Permeability Data: Deconvoluting the Ergun Papers
Abstract
In his 1952 publication, Ergun made the following proclamation: “Data of the present investigation and those presented earlier have been treated accordingly, and the coefficients and have been determined by the method of least squares. The values obtained are and, , representing 640 experiments.” In this paper, we demonstrate that because his experimental methodology was flawed, the corrected values, which his experimental results would otherwise have established for these coefficients, are significantly higher. This is, in part, because Ergun’s reporting of his measured data was ambiguous with respect to the embedded coefficients and . In addition, this ambiguity made it difficult for any subsequent researcher to figure out the true meaning of his empirical results which, in turn, resulted in his choice of the values for these coefficients being accepted by default in the scientific community.
1. Introduction
The Ergun equation is unquestionably the most popular equation in use at this time to represent the relationship between pressure drop and resultant fluid flow in packed beds, when the fluid profile contains significant contribution from kinetic effects. In addition, most engineering departments at major universities and third level educational institutions throughout the world offer a teaching module which has as its focus the now famous “Ergun equation.” The engineering literature, both chemical and otherwise, is replete with references to the equation and many of the studies reported therein have used it as a benchmark, against which to compare and contrast their empirical results. The equation was developed by Sabri Ergun in the late 1940s/early 1950s. It was derived, in part, from experimental measurements and, in part, from theoretical postulates. Ergun’s initial paper was published in 1949 in collaboration with Orning. In two followup papers published in 1951 and 1952, Ergun reported his empirical results of enormous amounts of experimental runs aimed, in part, at calibrating the equation in terms of its two builtin coefficients. In this paper, we review these papers on this subject matter which we refer to collectively as the “Ergun papers.”
The first empirical data on the relationship between fluid flow rate and pressure drop across a packed bed was reported by Darcy in 1856 [1]. The resultant Darcy law states that when the flow is streamline, the fluid flow rate through a packed bed is proportional to the pressure drop across the bed. The law has limitations, however, because it only applies to streamline flow and, in addition, tells us nothing about the factors which go into the constant of proportionality between fluid velocity and column pressure drop. Kozeny and Blake were the first to define in some greater detail the elements that define the constant of proportionality in Darcy flow [2, 3]. The KozenyBlake equation, subsequently modified by Carman to accommodate irregularly shaped particles, has been used extensively to evaluate streamline flow but the coefficient therein has been the source of much controversy over the past fiftyplus years [4–9].
Scheidegger, in his textbook at page 89, states: “It is obviously quite possible for two porous media of the same porosity to have entirely different permeabilities [10]. Thus, if a correlation function between the two quantities is sought after, it cannot be unique. Therefore, most empirical correlations contain some other factors, usually vaguely identified with alleged geometrical quantities. They are, however, nothing but undetermined factors used in order to make the data fit the desired equations. There are even a series of claims for “general” relationships, usually supposed to be true for “average” porous media, whatever that means.” He continues: “Thus, Fancher, Lewis and Barnes (1933, 1934) and Manegold (1937) observed that air permeabilities are higher than liquid permeabilities in the same porous medium as calculated from Darcy’s law. In fact, this is an indication that Darcy’s law is not valid for gases.”
In the textbook, Transport Phenomena, 1960 edition, the authors accept the value of 150 for the viscous coefficient in the KozenyBlake equation [11]. At page 199 of the book, which has become the gold standard at most chemical engineering schools, the authors state: “Analysis of a great deal of data has led to the value of 25/6, which we accept here.” They go on to qualify its use as being valid for column porosities less than 0.5 and insert this ratio into the equation to show that it translates into the value of 150 which would correspond to the value of in the Ergun equation. Curiously, however, they provide no reference for the “great deal of data” so there is no way for us to determine from whence it came. In the second edition of the text, published in 2002, at page 190 the authors further state that “it has been found that replacing 16 by 100/3 allows the tube bundle model to describe the packedcolumn data.” They go on to show that this choice of ratio, though different from the one quoted in the previous edition of the book, results in the same value of 150 for the viscous coefficient and, yet again, no reference given as a foundation.
Happel and Brenner, in their textbook state: “For packed beds of uniform spheres the CarmanKozeny equation within the range to , gives excellent correlation with a Kozeny constant of [12]. A recent study by Anderson, including additional sources, indicates that for uniform spheres . Anderson proposes a refinement in which is taken as a function of instead of being assumed constant. A large volume of data on beds consisting of a variety of nonspherical particles indicate that , independent of shape and porosity from to . As shown in Table 84.2, agreement of the CarmanKozeny relationship with hydrodynamic theory based on the free surface cell model is excellent.” It will be appreciated that a Kozeny constant of 5.0 corresponds to a value of 180 () for the coefficient in the KozenyBlake equation, as modified by Carman.
Stuart Winston Churchill, at page 524 of his textbook states: “Ergun combined the CarmanKozeny equation but with a coefficient of 150 instead of 180” [13]. He goes on to show a plot of Ergun’s data which he describes in the caption as “Comparison of experimental data for the pressure drop through packed beds of spheres, cylinders, sand, pulverized coke, and Raschig rings…from Ergun.”
In his handbook, Uwe Dieter Neue states: “As we know from experience, the backpressure of a column packed with small particles is larger than the backpressure of a column packed with large particles…The relationship is known as the KozenyCarman equation” [14]. This author goes on to identify the value of 185 as the coefficient in his version of the equation.
Dullien, in his textbook, at page 246 states: “MacDonald et al., have tested the Ergun equation using much more data than was ever used before by others [15]. The following relations were formed to give the best fit to all of the data.” Dullien, who was also a coauthor on the MacDonald paper, then points out that, for smooth particles, values of 180 and 1.8, for and , are best, but for roughest particles the value of 1.8 should be replaced by the value of 4.
Rhodes, in his textbook at page 85, shows the Ergun equation with the typical coefficients 150 and 1.75 but also includes the value of 180 for the viscous constant in what he describes as the CarmanKozeny equation [16].
Guiochon et al., in their textbook state at page 152: “It has been claimed sometimes that spherical particles have a larger permeability than irregularshaped ones, but definitive data are not available. The effect, as far as it exists, may in part result from uncertainties in the definition and measurement of the average size of irregular particles” [9].
Finally, Giddings, in his first textbook published in 1965, at page 209, displays in Table 5.31 arguably the most comprehensive data summary of packed column permeability in the published literature [7]. This data, when analyzed with respect to the form and function of the KozenyBlake equation, establishes a value of 267 for the viscous constant which corresponds to the value of in the Ergun equation [4]. In addition, at page 65 of his second textbook, published in 1991, the author displays what he refers to as the KozenyCarman equation with the value of 180 for the viscous coefficient. In a footnote below the equation, however, the author adds the parenthetic qualification “(This equation appears to work better for chromatographic materials if 180 is replaced by 270)” [8].
Ergun and Orning were the first to piece together an equation for flow in porous media which contained a viscous term and a kinetic term and in 1952 Ergun announced specific values for the equation coefficients, and . These values for the coefficients, however, have not been shown to be universally effective at correlating empirical values [17–19].
2. The 1949 Paper
We begin our discussion of the Ergun papers by going back to Ergun and Orning’s original 1949 paper, not only because it was the genesis of the Ergun equation, but also because it planted the seeds of error which eventually led Ergun to come up with his mistaken values for the equation residual coefficients, and , in 1952. Ergun and Orning begin with Kozeny’s equation for viscous flow, which they say was proposed by Carman for use with liquids, and extended for use with gases by Lea and Nurse. The authors then move to a discussion of packed columns under high flow rate conditions in which the pressure drop appears to vary with some power of the velocity, the exponent ranging between 1 and 2. They note that Blake postulated that this fluid behavior was analogous to that in open tubes and, thus, he proposed an equation which was based on a friction factor meant to normalize for the kinetic effects in packed beds of granular material. This friction factor, in turn, was a function of the particle based Reynolds number.
The authors next address the collective effects of the viscous and kinetic contributions to overall pressure drop, noting that there is a smooth transition from viscousdominated to kineticdominated effects. They argue, therefore, that there should be a continuous function relating pressure drop to flow rate. They conclude that “a general relationship may be developed using the Kozeny’s assumption that the granular bed is equivalent to a group of parallel and equalsized channels, such that the total internal surface and the free internal volume are equal to the total packing surface and the void volume, respectively, of the randomly packed bed.” The authors now turn to Poiseuille’s equation for flow through an open channel: where = pressure gradient across channel, = average interstitial fluid velocity, = diameter of capillary, and = fluid absolute viscosity.
Taking a term analogous to that introduced by Brillouin to represent the kinetic energy losses in a capillary, the authors add it to Poiseuille’s equation: where = fluid density.
At this point, Ergun and Orning postulate the number of channels per unit area and their size can be eliminated in favor of specific surface and fractional void volume and reexpress their equation as where = column external porosity, = particle specific surface area, and = fluid superficial linear velocity.
The authors point out that although this equation is similar to others which relate pressure drop to polynomials of the flow rate, it differs to the extent that the coefficients, that is, 2 and 1/8, have definite theoretical significance. Presumably, they conclude this because these values have their genesis in the physical relationships pertaining to cylindrical channels.
They further theorize that, for a packed bed, “the flow path is sinuous and the streamlines frequently converge and diverge. The kinetic losses, which occur only once for the capillary, occur with a frequency that is statistically related to the number of particles per unit length. For these reasons, a correction factor must be applied to each of the energy loss terms. These factors may be designated as and .” Accordingly, they reexpress (3) as follows: They point out that and may be determined experimentally. They claim that experiments with randomly packed materials of known specific surface area have led to the adoption of a value of 5 for , which is close to the theoretical value of (/2)^{2}, or 2.47, as proposed by Hitchcock [20].
The authors then assert that integration of (4), assuming isothermal expansion of an ideal gas, leads to the equation where = average superficial linear velocity of fluid (gas) at the mean pressure, = mass flow rate, and = column crosssectional area.
The authors state that Leva and Grummer reported that, in packed tubes, when the viscous forces are predominant, the pressure drop is proportional to , whereas, at higher flow rates, the corresponding dependence on the void fraction is , which is just what would be expected from (5) [21]. They then point out that a plot of versus would give a straight line, the intercept and slope leading to the values of and .
In Figure 1 of their 1949 paper, they display their own measurements in the form of what they say are “Typical plots of data obtained to test this dependence of pressure drop on gas flow rate.” However, the ordinate of the plot contains the velocity designated as (not ). This symbol appears nowhere else in the paper and is conspicuously absent from the glossary of terms at the end of the paper. We interpret this to mean, however, that the authors intentionally use interchangeably the superficial velocity at the mean pressure and at operating conditions of standard temperature and pressure STP. (This is also the same protocol used by Ergun in his 1952 paper. See his discussion in that publication of his equation (10) relative to his Figure 4.) Additionally, the coordinates of the plot have units of measure corresponding to a plot of on the ordinate against on the abscissa. The plot contains 5 different sets of measurements which exhibit straight lines with varying slope and intercept values. In their Figures 2 and 3, they show two additional plots in which they have taken data from the work of Burke and Plummer and Oman and Watson, respectively, which also give straight lines, thus allegedly validating (7) [22, 23]. (The ordinates of these plots have the velocity also designated as .) Additionally, in their Table 1, they provide values for and which range from 1.7 to 3.0 and from 2.4 to 4.5, respectively, which they derive from numerous data sets, including their own.
The authors state that, using the values of and from Table 1 and considering their (8), one can discern the relative importance of the viscosity and the kinetic terms at different Reynolds numbers. Specifically, they point out that, for Reynolds numbers around 60 and a column porosity of 0.35, the two have nearly equal effects upon pressure drop. However, for a Reynolds number of 0.1, viscosity accounts for 99.8% of the pressure drop, while, for a Reynolds number of 3000, kinetic effects account for 97% of the pressure drop. They go on to point out that because packed columns are generally operated in the range from Reynolds numbers of 1 to 10,000, both terms are essential in calculations of pressure drops.
They conclude their discussion on fixed beds with the statement that when using geometrically arranged packing, as opposed to random packing, (5) is valid except that the coefficients may vary markedly for different arrangements, even at the same fractional void volumes.
2.1. The Theoretical Parameters and
In describing the role of and , the authors say that “For a packed bed the flow path is sinuous and the streamlines frequently converge and diverge. The kinetic losses, which occur only once for the capillary, occur with a frequency that is statistically related to the number of particles per unit length. For these reasons, a correction factor must be applied to each term. These factors may be designated as and .”
This postulate raises more questions than answers. Assuming, for example, that the statement concerning the kinetic losses is correct, why does this necessitate a correction factor for the viscous term? After all, the adoption of Kozeny’s postulate for streamline flow regarding the equivalency of the granular bed to open channels supposedly captures all the flow considerations in the viscous term. Another obvious question relates to the rationale underlying the assertion that the kinetic losses “occur only once for the capillary.” Finally, since the authors abandoned any effort to relate pressure drop to the array of individual particles in a packed bed in favor of a theoretical model of equivalent channels, the concept of kinetic losses being related to “the number of particles per unit length” would appear to be out of place in this newly adopted “channel” model.
The authors attempt to justify their values for and in their (9), which includes an expression relating an entity, , which they call “the friction factor” to both and . We repeat that equation here: where the modified Reynolds number at STP.
(Note: the author’s (9) has a misprint. The which appears outside the bracket on the right hand side of the equation should be .)
Presumably, this is supposed to demonstrate that their values for and can be derived from empirical data but it is not superficially obvious where the value of 96 comes from or what it represents. Characteristically, the authors do not explain how they arrived at their (9) from their starting equation (7).
It follows, however, that their (9) cannot by itself lead to independent values of and since there are two unknowns and only one equation. In fact, their lone friction factor identified in their (9) is based on commingling of values associated with both the correlation coefficients and the independent variables in both the viscous and the kinetic terms of their equation. This commingling of the terms which make up the separate and distinct contributions of viscous and kinetic elements renders the resultant correlation coefficient (96) meaningless. In the end, therefore, both of the values of and must be derived from the linear plot in Figure 1 and cannot be derived from the lone friction factor which serves only to underscore the interdependence of the one upon the other. The values of and , therefore, are without specific meaning since they represent some unknown mixture of both viscous and kinetic contributions.
2.2. The Mean Pressure Postulate
In our discussions to this point the relationships between , , and are implicit. We use the term to represent the superficial velocity of the fluid when the fluid is a liquid and thus not compressible, whereas the symbol is used when the fluid is compressible as in the case of gases. These relationships are where = the density of the fluid at the mean pressure.
We may now express the ratio of and . Thus, It follows that The mass flow rate is related to the volumetric flow rate through the density of the fluid. Thus, we may write the following: It follows that where = the volumetric flow rate of the fluid.
We note from this relationship that there are an infinite number of combinations of and which will produce an equivalent value of .
The author’s theoretical postulate concerning the “mean pressure” is the equivalent of measuring the pressure drop at the inlet and outlet of the column and equating the average of the two values to the in (5). The in (5), however, is simply the difference between the pressure at the inlet and outlet of the column, not the average of the two. In addition, it represents the theoretical differential pressure drop since all other parameters in the equation are separately identified in the stated equation format. Let us, therefore, insert the theoretical boundary condition pertaining to the measured pressure drop, which is contained in their postulate of “mean pressure,” on the left hand side of (5). We accomplish this by substituting /2 into the equation for . Thus, this gives where .
We digress here to point out a potential problem in interpreting (12). Note that the left hand side of the equation has the velocity specified as that at the mean pressure, . This is a very specific velocity; it is separately identified with its own symbol and discrete value but exists only in the theoretical context of Ergun and Orning’s equation. The right hand side of (12), on the other hand, has the velocity embedded in the parameter , as we pointed out above. In addition, since a given absolute value of can have an infinite number of combinations of fluid velocity and fluid density, (12), as stated, is ambiguous with respect to velocity on the right hand side. Restating (12) in unambiguous terms requires a substitution for the mass flow rate, , to expose the embedded values of the velocity and density, as follows.
Substituting for gives Substituting for gives Isolating the theoretical “mean pressure” gradient on the left hand side, that is, multiplying across by the quantity , gives The expression, , on the left hand side of (15), is the theoretical equivalent to the measured “mean pressure” gradient. In the theoretical version both the mean pressure gradient, , and the fluid density, , are separately identified. In the measured value, however, the effect of the density is embedded in the measured value of the pressure gradient and is not separately identified and, of course, there is no 2. We note that this theoretical equation now establishes a relationship between measured pressure gradient at the mean pressure, on the left hand side of (15), and superficial velocity at STP, which involves only quantities which we can measure outside the column, on the right hand side of the equation. Thus, we can establish a correlation between measured mean pressure and superficial velocity at STP. Note, however, that this procedure embeds the density at STP, , into the viscous term as well as embedding the square of the density at STP, , into the kinetic term.
Normalizing for in (15) gives We can see then that this procedure requires us to adjust the theoretical pressure gradient at the mean pressure, that is, the left hand side of the equation, that is, , by the ratio of the density at the mean pressure to the density at STP (/). This is necessitated because our reference superficial velocity on the right hand side of (16) is reflective of the fluid density at STP, , when the flow rates were recorded. Therefore, we must “back out” the bias in the measured pressure gradient on the left hand side of the equation by an amount equal to the effect of the embedded ratio of the densities at the mean pressure and STP, in order to maintain parity between the two scenarios.
Equation (16) now represents the properly adjusted theoretical pressure/flow equation, based upon the author’s mean pressure postulate, which allows us to correlate the measured pressure gradient of a compressible fluid to that of the volumetric velocity measured at STP at the outlet of the column. In fact, this equation format is valid whatever the reference density that one chooses to measure the volumetric flow rate at assuming of course that the density at STP, , is replaced by the appropriate reference density . Note, however, that this postulate has embedded a value of 2 into the right hand side of the equation. This means that author’s postulate concerning the mean pressure has an embedded assumption that the ratio of the actual superficial velocity inside the column to that at STP is 2. We can show this mathematically as follows: Substituting for in (16) gives Dividing across by 2 gives We can see then that (19) is a valid relationship between pressure gradient and the other independent variables whenever all variables are taken within the same frame of reference. In other words, when applied to a packed granular bed all parameters must be measured either inside the column or outside the column. Of course the measurement of pressure gradient outside the column does not make much sense and, therefore, this equation is only valid when the fluid is incompressible as in the case of a liquid where the density parameter is a constant value regardless of whether it is measured inside or outside the column. Accordingly, when using liquids it is customary to measure the pressure inside the column and the flow rate outside the column.
Thus, we restate (19) in order to capture the effect of fluid compressibility in the measured pressure drop as follows: Equation (20) is the correct equation format to be used when a compressible gas is used to generate pressure inside a column and the mass flow rate is measured outside the column. This format properly identifies the measured pressure drop and also the fact that the effect of fluid compressibility is embedded within the measured pressure drop. Therefore, (19) and (20) are identical representations of the same relationship with the proviso that the former represents the theoretical pressure drop (nothing unaccounted for embedded) but the latter represents the measured pressure drop which has embedded quantities which are not accounted for on the right hand side of the equation. Accordingly, (20) is not overtly balanced.
3. The 1951 Paper
Ergun, this time as the sole author in 1951, begins by noting that an exact knowledge of particle solid density is important for determining the void volume fraction (external porosity) in a column comprised of porous particles. This is because, without a direct independent method to measure the void volume fraction of a packed column, the usual practice is to measure the mass of particles in the packed column which enables the calculation of the column packing density. Using the column packing density, in turn, enables the calculation of the void volume fraction if the particle solid density is known. Unlike nonporous solids, however, the particle density of porous solids is not equivalent to the particle solid density and is a function of the particle size. The particle density of porous particles always increases as the particle size is decreased and becomes equivalent to the particle solid density when the particles are reduced to being nonporous. This results from the fact that as a given particle size of porous solids is further broken down to smaller particle sizes, some of the pores in the material are eliminated. Thus, in order to determine the particle solid density one must reduce the particle size to that of the smallest size of the pores within the particle. It follows that the particles may then be considered nonporous. This same principle also applies to the determination of the specific surface area of any porous particles.
Ergun continues that, because of the lack of a successful method to determine directly the particle solid density of porous particles, it has been generally the practice to assign an apparent particle density and use empirical adjustment factors such as shape and surface factors to correct for this lack of knowledge of the particle solid density. He says that a method to circumvent this problem is to consider a porous particle as a nonporous particle surrounded by an impervious volume envelope. In this scenario, therefore, the particle solid density is then the mass of the particles divided by the impervious volume envelope. The mass of the particles is easily determined but the impervious volume envelope is not. If, however, gas is made to flow through a packed column of porous particles, the gas will fill the pores in the particles. Thus, when the mass of the gas flow rate is measured at the exit of the column, it will be representative of flow only in the spaces between the particles because the mass of gas in the pores remains constant and stagnant. Further, the mass of the flowing gas will not be influenced by the shape of the particles as the gas will flow in the interstitial pore space whatever its shape happens to be. Therefore, the mass flow rate of the gas will be representative of the “impervious volume envelope” in the model.
3.1. Packing of Porous and Nonporous Irregularly Shaped Particles
Ergun next proceeds to test the process by carrying out experiments. The description that he gives of his experimental apparatus is minimal at best. He reports that “The apparatus consisted of a glass tube, two capillary flowmeters, and three manometers.” He proceeds to describe in detail the technique by which he packed the glass tube with particles and the position of the manometers within the column which was the method he used to measure the pressure drops. However, he is silent on how he used the capillary flowmeters to record gas flow rate. Since it is not possible to position the capillary flowmeters inside the column, the only place that his equation is valid, they must have been deployed outside the column where his equation is not valid. It is, therefore, surprising that he provides no details on what flow parameter (mass or volumetric) or what technique he used to capture the flow rate information outside the column and the basis upon which he related such measurements to the pressure drop measurements taken inside the column.
His column packing technique involved the packing of a known mass of particles of narrow particle size distribution by using an upward flow of gas to expand the packed bed. A vibrator was also used to agitate the particles while forming the packed bed structure. The flow rate of the upflow of gas will determine the degree of bed expansion and thus is an efficient way to vary the external porosity of the column. When the vibrator and gas upflow are turned off the bed formed has a fixed porosity. The measurements of pressure drop and gas flow rate were taken in the opposite direction of gas flow, that is, in a downward direction. This prevented the bed from expanding under the flow of gas and thus all measurements represented that of a fixed bed. Each particle size fraction was packed to between 6 and 12 different external porosities. Ergun then compares the results for calculated solid particle density using both versions of the equations with both porous and nonporous particles. In addition, he compares his results for his nonporous particles to conventional wet immersion techniques and concludes that they compare favorably.
Ergun’s data consists of three distinct categories of measurements. They are as follows.(a)Table I: Particle Densities of Crushed Nonporous Solids. In this category of data, Ergun shows his measurements and solid particle density determinations of 4 different particle size fractions of pulverized glass and river sand. He also displays the results obtained by the conventional technique of water displacement. We note that, in all cases, the values obtained by the water displacement technique correspond well to the values obtained by Ergun via his (5). Ergun’s Figure 1 displays his data for the 80–100 mesh Brown River Sand contained in this category of particle definition (nonporous solids).(b)Table II: Particle Density Determinations for High Temperature Oven Coke. This category of data contains the results of measurements of nine different particle size fractions of high temperature oven coke which represents columns packed with porous particles. Ergun’s Figure 2 contains data from this category having a particle size distribution of 8–16 mesh.(c)Table: IV Particle Density Determinations of Cokes of 40 to 60Mesh Size. This category of data is again all porous particles but this time having the same particle size fraction of 40–60 mesh but having a different origin and thus various solid particle densities. Ergun’s Figure 3 data is taken from this category but does not specify which individual coke designation it represents.In this paper, Ergun deals with both nonporous and porous particles. His experimental protocol is designed to develop a methodology to identify the external porosity of a packed column of porous particles. The external porosity of a packed column, represented by in our definition of terms, is the volume of the free space between the particles expressed as a fraction of the total volume of free space in the empty column. Because, however, one cannot distinguish between the free space between the particles and the free space within the pores of porous particles, a determination of the external porosity is difficult in columns packed with porous particles.
Ergun is attempting to exploit his newly developed pressure drop/fluid flow equation by backcalculating for the single unknown column external porosity. His methodology is based upon the principle that if one knows all the variables in the equation except one and one knows the product of all the variables, that is, the pressure drop, one can backcalculate for the unknown parameter. Accordingly, he first attempts to validate his concept by using nonporous particles of known solid particle density and comparing his results to the other independent method by which the original result was obtained. This information is contained in his Table I.
3.2. Contradictory Statements/Definitions
Ergun uses his 1949 equation developed with Orning as the starting point for his methodology. However, he introduces it in this paper (1951) as his (1) in which he defines the velocity term, , as “the linear gas velocity based on the cross section of the empty column.” This definition we recognize simply as the superficial velocity of the column based upon the usual differential pressure drop across the column, . This newly minted definition is in direct contradiction to his definition of the velocity term in his original development in which it was defined by himself and Orning as , which they defined as “the superficial velocity at the mean pressure.” Accordingly, it appears that Ergun is now overtly equating the superficial velocity at the recorded pressure drop across the column, , to the superficial velocity of the average (mean) of the pressure at the entrance and exit of the column, .
Moreover, in commenting on his experimental results in his Figures 1 and 2 in the main body of his paper, Ergun states: “When the ratios of pressure gradient to average velocity are plotted against mass flow rate according to (1), a straight line is obtained (his Figures 1 and 2) having intercept and slope .” The problem here is that his Figures 1 and 2 have their abscissa terms labeled as “flow rate at STP” with the designated units of cc./sec. which is a unit of volumetric flow rate not mass flow rate. It would appear then that Ergun is using some hodgepodge of parameters and definitions which would appear to confuse mass flow rate with volumetric flow rate, on the one hand, and superficial velocity at the mean pressure with superficial velocity at STP, on the other hand.
3.3. The Mysterious Multiplier (5.79 × 10^{3})
In the caption underlying his Figures 1 and 2 of the 1951 paper, Ergun asserts that a multiplier of 5,790 must be used to connect his data displayed in the figures to his data listed in his Table I. However, he is silent on the reason for such an enormous correlation multiple and does not give even a scintilla of hint as to what it represents. Undoubtedly, the value of 5,790 results from the exact experimental protocol which he was using to measure the flow properties of the fluid in his experiments. Characteristically, though, he is silent on the details of how and what he measured. For instance, he states that he used “two capillary flowmeters, and three manometers.” We wonder why the need for two capillary flowmeters. Perhaps one was for volumetric data and one for gravimetric? He describes the reason for using two manometers, that is, at two different bed heights within the column, but we wonder what the third manometer was for! Perhaps it was for measuring the pressure outside the column? Regardless, however, of what particular combinations of experimental apparatus and protocols Ergun used to define his multiplier, it is certain that they were aimed at accounting for the effect of fluid (gas) density, a topic which Ergun was careful to avoid in the main body of his paper. In fact, there is not even a single mention of the need to consider the compressibility of the fluid (gas) in this entire paper of 1951.
We have determined that the mysterious multiplier, = 5,790, was the ratio of the slopes of the straight lines obtained in the plots of his measured data. Ergun was, apparently, measuring both volumetric flow rate and mass flow rate at the column exit and by taking the ratio of the slopes of the straight lines obtained he generated his mysterious multiplier of 5790 which he could use to relate the two results. This explains the need for the two capillary flowmeters. Moreover, although Ergun was measuring both volumetric and mass flow rate, he chose to report only the results of his mass flow rate measurements in the many tables of this 1951 paper. In other words, he was measuring the equation of the line generated by his theoretical equation for volumetric flow rate which has the format, , and he was also measuring the mass flow rate, which has the format, , but he chose to report only the latter, his values for mass flow rate. In addition, he also used the same symbols, and , to represent the intercept and slope of the line, in either case, once in the 1949 paper when he was referring to the mass flow rate and once in his 1951 paper when he was referring to the volumetric flow rate. Although both equations of the line have the same value of intercept, , they do not have the same slope . Since, however, he presented his measured data in graphical form as volumetric flow rate in this 1951 paper (see his Figure 1), which has volumetric flow rate at STP on the abscissa, and reported his results as mass flow rate in the tables, he had to use this mysterious multiplier as a conversion factor between the two formats. Accordingly, we know that the mysterious multiplier is the ratio of the column crosssectional area to the density of the gas at STP, .
4. The 1952 Paper
In his 1952 paper, Ergun builds upon developments in his two previous papers. He begins his 1952 paper with the postulate that “pressure losses are caused by simultaneous kinetic and viscous energy losses” and concludes that “the following comprehensive equation is applicable to all types of flow.” Consider where = pressure loss in force units, = gravitational constant, = fractional void volume in bed, = absolute viscosity of fluid, = superficial fluid velocity measured at average pressure, = mass flow rate of fluid, = height of bed, and = average spherical diameter equivalent of particle.
Ergun states that the purpose of this paper is to verify empirically an earlier theoretical development and that the new experimental work involved gas flow through crushed porous solids. He underscores the fact that this is in contrast to many previous studies in which the experiments involved nonporous solids. He goes on to say that most authorities agree that the most important factors to be considered are(1)rate of fluid flow,(2)viscosity and density of fluid,(3)closeness and orientation of packing,(4)size, shape, and surface of particles.Ergun notes that a linear relationship exists between , the viscous friction factor, and , which is otherwise known as the modified Reynolds number. Utilizing a fitting algorithm of least squares to his measured data, he determines from 640 experiments the empirical values of and . The data set involved various sized spheres, sand, pulverized coke, and the following gases: CO_{2}, N_{2}, CH_{4}, and H_{2}. Thus, the particles used in the study involved both porous and nonporous solids, whereas the fluids used in all cases were compressible gases. The values for and which Ergun claims were generated by his experiments are 150 and 1.75, respectively. Substituting 150 and 1.75 for the values of and , respectively, Ergun goes on to express his equation as This is a representation in which the viscous energy losses have been normalized.
He suggests that another form of representing the equation is This is a representation in which the kinetic energy losses have been normalized.
4.1. Ergun’s Presentation of His Measured Data
Figure 1 in this 1952 paper is a plot of versus . In this graphical representation of his measured data, Ergun focuses on the linearity of the relationship. Although the quantities plotted on both axes supposedly represent his measured values and are not dimensionless, Ergun does not specify the units of measure. This is a critical omission, especially in a paper designed to validate a theoretical model. Figure 2 in his paper contains two plots which show the influence of column porosity in both viscous and kinetic terms of his equation. Ergun points out that since both plots are straight lines which pass through the origin, this demonstrates that the respective porosity dependence terms for the viscous and kinetic energy losses are correct. In his Figure 3, Ergun simply plots his results from additional experiments carried out with a different particle size and with different bed porosities. Unlike his data in his Figure 1, however, he does not provide the underlying packed column densities, which are necessary to independently evaluate this plot. The fact, though, that the plot is also a straight line which passes through the origin suggests that the viscous porosity dependence term is universal in nature. The plot in his Figure 4 is a different representation of Ergun’s measured data than that displayed in his Figure 1. It is unclear exactly what the significance of this plot is, other than condensing into one plot both of his plots in his Figure 2, since it results in values for the intercept and slope identical to those derived in the plots in his Figure 2. Again, no units of measure are specified.
Thus, Ergun made a significant contribution to the understanding of the pressure/flow relationship in granular beds by recognizing the additive nature of viscous and kinetic energy losses to the overall pressure drop. This advancement in understanding enabled him to establish two versions of a friction factor, each of which could be used for normalizing either kinetic or viscous energy losses.
5. Anatomy of the Ergun Equation
The Ergun equation is based upon a basic principle annunciated by Ergun and Orning in their 1949 publication which they expressed as follows: “The ratio of pressure gradient to superficial fluid velocity in packed columns is shown to be a linear function of fluid mass flow rate.” Based upon this a priori notion, they developed their equation which established a relationship between fluid flow rate and differential pressure across a packed column. We repeat here that equation from their 1949 paper: The authors embedded into the equation the term on the viscous side and the term on the kinetic side based upon a theoretical postulate which arose out of two concepts. One of the concepts was based upon the notion of cylindrical channels which contributed the 2 to the viscous term and the 1/8 to the kinetic term. The other concept had to do with the notion of sinuous flow between the particles in the packed column and contributed the to the viscous side of the equation and the to the kinetic side. Since we do not accept the rationale underlying either concept for the reasons we pointed out above, in the first instance, and since it complicates our uncovering of the correct values for the coefficients and , in the second instance, we will now undo these theoretical developments and restate the equation by recombining these terms into the residual coefficients and . Thus we may write Next, we substitute for and , which gives Recombining the 36 into and the 6 into gives where and are the residual components of viscous and kinetic contributions, respectively, embedded in the measured instantaneous pressure drop, , which Ergun and Orning’s theoretical development did not account for in their final equation. We point out, however, that there is nothing fundamentally “constant” about the residual coefficients and . They are simply the remainder of the unaccounted for contributions, but assuming that the equation is valid in the first place, the value of this remainder should always be the same after all variables are taken into account, and, thus, it is in this context that we refer to them as being “constant.”
5.1. The 1949 Columns with Spherical Nonporous Particles
In the 1949 paper, Ergun and Orning introduced their newly minted equation along with a significant amount of empirical data. They present their results in graphical form in three plots of data for packed columns which are linear, Figures 1, 2, and 3. We note that they measured the flow rate of the fluid outside the column. “Gas flow was measured with a series of capillary flowmeters, each calibrated against wettest meters.” We therefore know that the values which they used for mass flow rate displayed on the abscissa of the graphs, at least in their own plots, are representative of measurements taken at standard temperature and pressure (STP).
On the other hand, we know that they measured the pressure drops inside the column. “Pressure drop measurements were made in a glass tube, 1 inch in inside diameter and 30 inches long, fitted with a frittedglass disc to obtain uniform gas flow at the bottom and with pressure taps at the top and at a point just above the frittedglass disc.” Conspicuously absent from their measurements, however, is any data or mention of the independent variable of “superficial velocity at the mean pressure” which appears in their equation. Additionally, their description of the experimental apparatus used gives no hint of how one would even measure this parameter. It is also significant that, instead of reporting their measurements of all independent variables on an individual variable basis, a customary practice, the authors made only a partial reporting in the form of their pressure drop and flow rate measurements displayed in their Figure 1. Since their Figure 1 plot, however, gives only the ratio of pressure drop and velocity multiplied by column length, it is ambiguous with respect to the embedded values of and because the values of all the equation variables have not been identified.
We conclude, therefore, that Ergun/Orning measured the pressure generated inside a packed column while passing a fluid (gas) through the column. A gas is compressible and therefore its density is a function of the pressure at which it is measured. Even if one measures the mass flow rate outside the column, however, one still has to know the density of the fluid inside the column in order to calculate the volumetric flow rate inside the column, because it is this volumetric flow rate that underlies the calculation of superficial fluid velocity which, in turn, is called for in both the viscous and kinetic terms of the authors’ pressure/flow equation.
5.2. A Closer Look at the Experimental Data
We begin our analysis by generating our own spreadsheet of calculations underlying author’s Figure 1. The data underlying their other plots, for example, Figures 2 and 3, were taken from the literature and they did not report all the appropriate underlying independent variables in the paper, so we neglect these data sets in our analysis.
To begin, we define here the basis upon which we derive the values of and which appears in author’s Table 1. We can show our methodology by first looking to authors’ (7): It follows from the plot in authors’ Figure 1 which is a plot of the left hand side of this equation versus the quantity on the right hand side that the values of and are determined thusly: where [Intercept] and [Slope] represent the intercept and slope of the line in their Figure 1, respectively. These calculations are displayed in our Table 1 for the data corresponding to the columns in the author’s Figure 1 which were tested with nitrogen: lead shot with particle size of 0.497 mm (our column A), glass beads with particle size 0.570 mm (our column B), and, finally, lead shot with particle size 0.562 mm (our column C). The particles in these columns were totally smooth, spherical in shape, and nonporous, exactly what would be generally accepted in the field as “known standards.” This results from the fact that the physical characteristics of these particles can be well characterized by various independent techniques and there exists no doubt about either surface area or solid particle density, two physical properties which are necessary in Ergun’s calculations underlying his equation. It will be appreciated that Ergun did not measure the particle size as one might do in these days of sophisticated analytical instrumentation. Rather, he measured the surface area of the particles () by nitrogen absorption, or the equivalent, and calculated the average particle size as described in his paper (). In addition, the value of the solid particle density is used to calculate the column porosity, , another parameter specified in the equation.
