Quinn’s Law of Fluid Dynamics: Supplement #3 A Unique Solution to the Navier-Stokes Equation for Fluid Flow in Closed Conduits

The recent publication of Quinn’s Law of Fluid Dynamics brings into focus longstanding contradictions regarding permeability in closed conduits that have littered the fluid dynamics landscape for more than 150 years. In this paper, we will use this new level of understanding to explain these contradictions, in layman’s terms, and resolve them, by introducing for the first time, as far as we know, a unique solution to the Navier-Stokes equation for fluid flow in closed conduits, which is understandable by knowledgeable physicists, engineers, chromatographers and aerospace enthusiasts alike, but who may not necessarily be versed in the abstract jargon of a graduate in advanced mathematics. In addition, we will apply our unique solution to chosen illustrative worked examples, as well as those of third parties from the published literature. In so doing, we will demonstrate the utility of our solution, not only, to packed conduits containing particles having solid skeletons, but also, to empty conduits, which in the context of this new understanding of fluid dynamics in closed conduits, represents a special case of a packed conduit in which the particles are fully porous, i.e., they are made entirely of free space.


Introduction
Let us begin at the beginning. Most practitioners will say that the first real attempt to characterize the flow of fluids through closed conduits was made by Poiseuille circa 1846 [1], in the case of empty conduits, and by Henry Darcy in 1856 [2], in the case of conduits packed with solid obstacles. The former's work led to what is known today as Poiseuille's Law [3] and the latter's work to what is known as Darcy's Law [4]. Both these Laws teach that there is a linear relationship between fluid flow rate and the pressure drop across a given conduit. As time progressed, however, it became obvious that both these Laws had limitations, and in the intervening 170 years approximately, much effort has been devoted to ascertaining the underlying reasons [5][6][7][8][9][10][11][12]. Unfortunately, among practitioners, even to this day, there is much controversy regarding the parameters which constitute the pressure/flow relationship [13].
Darcy's original methodology used screened river sand packed into large pipes, through which he pumped water, and recorded the pressure drop across the packed pipe for each measurement of flow rate of the water [14]. It would be logical to conclude, therefore, that the precise nature of the sand particles, and the manner in which they were forced together inside the packed conduit, should be fertile ground for experimentation in any efforts to better understand the pressure/flow relationship in packed conduits [15]. Throughout the middle to end of the 20 th century, Blake, Kozeny, Carman, Bird Stewart and Lightfoot, Giddings, Halasz and Guiochon are just some of the more prominent scientists who devoted considerable effort to the elucidation of a fluid flow model capable of describing accurately the elements underlying this linear pressure/flow relationship, i.e., that portion of the fluid flow regime where viscous contributions to pressure drop dominate over kinetic contributions [16][17][18][19][20][21][22]. Unfortunately, however, their efforts did not culminate in a consensus of opinion and we are left today with the glaring contradictions regarding the value of the constant in the Kozeny/Carman equation, which is generally accepted as the most popular equation to describe diagram" which has been used as a popular "look up" chart by the engineering discipline for fluid flow designs which require knowledge of fluid flow in the region where kinetic contributions dominate [40]. The diagram is based upon the concept of "friction factor" which is a man-made entity however, being as it is a mathematical construct and, thus, unfortunately, suffers from the defects of its qualities [41]. It has, in addition, been updated from time to time since its original creation [42]. In more modern times, the Princeton Super Pipe has gained in popularity with respect to the theory of fluid dynamics in smooth pipes and has been credited with an update to the conventional concept of the "Law of the Wall" [43].
It is apparent from the brief history of the development of fluid flow theory outlined above, that empty and packed conduits formed separate and distinct categories of investigative effort involving fluid flow theory throughout the past 150 plus years. While some attempts were made to produce a unified fluid flow model which would seamlessly embrace both types of fluid flow embodiments throughout that period, none were successful, at least up until now [44]. With the advent of the Quinn Fluid Flow Model (QFFM) published last year (2019), this is no longer the case [45]. Accordingly, this paper is dedicated to elucidating a unified methodology for both packed and empty conduits, which a typical practitioner can take advantage of, whether that practitioner is an engineer, chromatographer or aerospace enthusiast.
We begin by introducing a methodology used in engineering circles called Hydraulic Conductivity [46]. To put this term in context for all disciplines, we show the relationship between hydraulic conductivity (∆H/L) and pressure gradient (∆P/L): We can see from the right hand side of equation (1) that hydraulic conductivity involves, not only, the pressure gradient across a packed conduit, but also, includes the fluid density, and the acceleration due to gravity. Thus, from an empirical perspective, a practitioner need only measure the pressure drop at any given flow rate, the length of the conduit, and obtain from reference text books the value for the density of the fluid used in the measurement, as well as the acceleration due to gravity. In addition, since it is customary when carrying out permeability determinations in packed conduits, to record the measured flow rate corresponding to the measured pressure drop, as fluid flux through the packed conduit, plotting fluid flux versus hydraulic conductivity is a popular engineering methodology. Thus, we can write: Where equation (2) represents the fluid flux, also called linear superficial fluid velocity, q=volumetric fluid flow rate, D=the conduit diameter.
Accordingly, in order to use the fluid flux parameter, the Fluid Mechanics 2020; 6(2): 30-50 32 practitioner must measure, in addition to the fluid volumetric flow rate, the conduit diameter.

Methods
When reporting empirical results of permeability in packed conduits, the Forchheimer fluid flow model is a popular engineering methodology, especially when the fluid flow regime involves significant kinetic contributions [47]. We can write the Forchheimer equation as follows: Normalizing equation (3) by dividing across by µ s , gives: Where, a, and b, are the Forchheimer coefficients for the viscous and kinetic contributions, respectively.
Thus, we can see from equation (3) that hydraulic conductivity is a quadratic function of fluid flux. It is customary in engineering circles to make a plot of equation (3), a typical example of which is shown in Figure 1.   As shown in Figure 1, the second order polynomial trend line associated with this plot renders the values of a, and b, both of which are represented as having a constant value, over all flow rate ranges.
Alternatively, equation (4) is often used by practitioners, since a linear trend line of this plot of normalized hydraulic conductivity versus fluid flux µ s , will render a straight line whose intercept represents the value of a, and whose slope represents the value of b, as shown in Figure 2.
As shown in Figure 2, this technique also generates a constant value for both coefficients.

Quinn's Law of Fluid Dynamics
Since Quinn's Law is universal, it applies to all fluid flow regimes. Specifically, it provides a detailed analytical definition of the fluid flow parameters which make up the numerical values of a, and b, for any experiment under study. Thus, we provide, herein, the definition for a, and b, as taught by Quinn's Law: In addition, Quinn's Law also provides all the needed relationships between the measurable fluid flow embodiment parameters necessary to completely establish the fluid flow relationship. Thus, we include some additional pertinent equations from Quinn's law: λ=(1+W N ) (11) where, r h =fluid drag normalization coefficient, i.e., 4, η=the fluid absolute viscosity, d p =the spherical particle diameter equivalent, d c =the diameter of the hypothetical Q-channel, n p =the number of particles in the packed conduit under study, and λ=the normalization coefficient for wall effects, generally equal to unity for packed conduits.
Since the parameter λ, however, involves a very complex definition involving many different independent and dependent variables, we refer the reader to the original publication of Quinn's Law for all the details concerning the components of λ [48]. Additionally, for the reader's convenience, we have included in the addendum to this paper, a comprehensive reference guide which provides, nomenclature, glossary of terms and all formulae from the original publication of Quinn's Law.

Solving the Navier-Stokes Equation for Closed Conduits
Quinn's Law is the only extant theory of fluid dynamics which includes an equation capable of describing the relationship between fluid flow rate and pressure drop that is unique in its ability to describe, accurately and precisely, this relationship throughout the entire fluid flow regime, including all three so-called regions of laminar, transitional and fully turbulent. We shall now explore in detail how this is accomplished.

Formula
The formula P Q =k 1 + k 2 C Q , which is the dimensionless manifestation of Quinn's Law, is a unique formula which combines the above identified variables in a manner never before contemplated.
3. Underlying theory The theory which underpins Quinn's Law is vitally important in executing the solution to the Navier-Stokes relationship, because of the important insights derived based upon an understanding of the boundary conditions embedded within the theory. Accordingly, what makes Quinn's Law unique is that it contains many parameters not identified in other fluid dynamic models, i.e., r h , k 1 , k 2 , β 0 , τ, λ, Q N , C Q , etc., etc., and, in addition, combines all the parameters in a unique arrangement not heretofore available in any other fluid model.
Thus, when the fluid flow rate, pressure drop and conduit diameter are determined by experiment and, accordingly, the Forchheimer values of a, and b, are known, based upon accurate measurements of these three variables over a broad range of flow rates, including the non-linear region, where kinetic contributions to measured pressure drop are significant and, in combination with the fluid property of kinematic viscosity, we can solve the N-S equation using Quinn's Law. Thus, we proceed as follows: It follows from equation (5) above that we may write: Similarly, it follows from equation (6) above that we may write: Therefore, in order to solve the N-S equation we must satisfy both equations (12) and (13) simultaneously.
From equation (12), let us assume that: From equation (13), let us assume that: Let us further assume that: x=αβ (16) Similarly, let us assume that: It follows that we may now write the solution to the N-S equation for closed conduits as: The above simultaneous solution for the values of d c and δ depends, not only, upon the independent variables identified above, but also, upon the value of λ in equation (13). However, λ, in turn, depends upon the value of other variables including d c , a dependent variable itself and, accordingly, and problematically, this is the conundrum of solving the N-S equation. Furthermore, the variable ε 0 (conduit external porosity), is clearly the most important variable amongst all the variables in the pressure flow relationship, since it appears in both the Forchheimer coefficients a, and b (equations (5 and (6)), and is also present in equations (18) and (19). Additionally, there is no more sensitive relationship in all of physics between the value of ε 0 in the Forchheimer coefficient b, and the value of the pressure gradient ∆P/L, when the fluid flow profile contains significant kinetic contributions.
Therefore, Quinn's Law is the only fluid flow model in existence today that can return a valid analytical solution, based upon pressure drop and flow rate measurements, for the values of d c and δ, simultaneously, for any given flow rate, in any given experiment, in a closed conduit, regardless of whether that conduit is packed with particles or empty, and regardless of where in the fluid flow regime that flow rate may fall, laminar, transitional or fully turbulent. This unequivocal assertion is a manifestation of the solution to the N-S equation for fluid flow in closed conduits.

Executing the Navier-Stokes Solution Using Quinn's Law
The theoretical basis of Quinn's Law, i.e. the Quinn Fluid Flow Model (QFFM), however, provides the means by which one can overcome this conundrum in solving the N-S equation for closed conduits, by establishing an understanding of the value of λ under three distinct closed conduit milieus, two of which are self-evident, based upon the underlying theory, and one of which necessitates an additional measurement of an independent variable.
We digress here to emphasize that the parameter λ, as defined in the QFFM, is bounded on the lower side by an asymptote which tends to the value of unity when the packed conduit tortuosity term, tau, is very large. The definition of the conduit tortuosity term tau, in turn, is based upon the architectural makeup of the particular closed conduit under study. It is, therefore, a totally novel concept for this parameter, amongst all other existing and competing theories of fluid dynamics. Consequently, the concept of conduit tortuosity, as defined in the QFFM, is what differentiates the fluid dynamics within different closed conduits and, accordingly, defines the three distinct closed conduit milieus regarding the values of λ which arise in the case of fluid flow in packed and empty closed conduits. These distinct milieus may be catalogued as follows: 1) Packed conduit with low value of the ratio D/d p The most general case of a packed conduit is when the ratio of D/d p is low, say less than 10. In this scenario, Quinn's Law teaches that there is a significant primary wall effect at very low values of the modified Reynolds number and, consequently, the value of λ will not be exactly equal to 1.0 and, together with the value of the Forchheimer coefficient b, will vary based upon the fluid velocity used in any experiment under study.
It is important to understand, however, that in this milieu, even though the primary wall effect is significant at very low values of the modified Reynolds number, it will only manifest in the pressure gradient measurements at moderate values of the modified Reynolds number. This is because all wall effects, and therefore the λ parameter, manifests only in the kinetic term of Quinn's Law, which has a relatively small contribution to the measured pressure gradient at very low values of the modified Reynolds number. One could, theoretically, make measurements at very high modified Reynolds number values, where the value of λ would tend to unity as the boundary layer is dissipated, but this could require large pressure drops, not practical in a typical practitioner's laboratory. Therefore, alternatively, one must know, in addition to the values of the Forchheimer coefficients a, and b, at least one more independent variable, to solve the N-S equation in this milieu scenario at reasonable pressure drops. That independent Equation for Fluid Flow in Closed Conduits variable is n p , the number of particles in any packed conduit under study, which must be independently measured in this scenario of a packed conduit. Again, we point out that in this scenario, the plotting techniques using equation (3) and (4) will not produce accurate values for the Forchheimer coefficients a, and b.
Thus, Quinn's Law is the only fluid flow model capable of validating the accuracy and precision of the underlying conduit variables of d p , and ε 0 , based upon Forchheimer type measurements of fluid flux and hydraulic conductivity, and which also include an independent measurement of the value of n p , in this milieu, where the primary wall effect is significant at low to moderate flow rates.
2) Packed conduit with large value of the ratio D/d p A packed conduit where the ratio of D/d p is large, say greater than 30. This represents a special boundary condition in the application of our solution. In this scenario, the QFFM teaches that there is no wall effect of any kind, either primary or secondary, and, consequently, because the value of the packed conduit tortuosity term, τ=δγ, is very large, the value of λ 1, at all fluid velocities across the fluid flow regime, i.e. all values of the modified Reynolds number, R em . Accordingly, the measured values for the Forchheimer coefficients a, and b, will both be constant at all fluid velocities. Thus, the above solution for the values d c and δ is absolute at a value of λ=1, since in equation (13), all the values of the parameters are uniquely defined at this boundary condition. This is typically the case for most commercially available packed conduits, since they are typically designed with large ratios of D/d p for performance related reasons. In this scenario, the technique outlined above of identifying the values of the Forchheimer coefficients a, and b, by plotting equations (3) and (4) will yield accurate values across the full spectrum of modified Reynolds number, if sufficient flow rate measurements are taken both in the linear and non-linear regions of the flow regime, which includes the region in which kinetic contributions to the measured pressure drop are significant. Thus, in this new teaching, measurements which include kinetic contributions are critical to identify an accurate value of the Forchheimer coefficients a, and b, regardless of what flow regime a particular experimental protocol may be focused, i.e., permeability studies in the laminar regime are included in this qualification.
We emphasize that in the literature for packed conduits in many applications, and especially in the field of chromatography, kinetic contributions to measured pressure drop have been totally ignored in favor of just doing measurements in the laminar regime. This results in inaccurate values for the Forchheimer coefficients a, and b, which, in turn, means that studies carried out under this set of experimental protocols will not facilitate validation of any underlying packed conduit variables. Indeed, when other fluid flow models are used in this scenario, such as the popular Kozeny-Carman model, in the case of packed conduits containing solid particles, and the Hagen-Poiseuille model, in the case of empty conduits, to back-calculate either the value of D, d p or ε 0 , they will provide only crude estimates of the true value of these parameters.

3) Empty conduits
All empty conduits regardless of the independent variable values which define them, i.e., the conduit diameter, D, the conduit length L, or the inner conduit wall roughness k, represent another special case of a packed conduit in the QFFM. Thus, although Quinn's Law teaches that an empty conduit has a relatively low tortuosity value and, consequently, a large value for λ at low flow rates, this is offset by the fact that its value is always constant, i.e. τ=3/16, which is a consequence of four limiting boundary conditions for an empty conduit: (a) d p =D, (b) δ=1/8, (c) n p =-n pq , and (d) ε p =1, where ε p represents the particle porosity (not to be confused with ε i , the conduit internal porosity), and when its value is unity, as in the case of an empty conduit, represents particles of free space. Thus, in an empty conduit, which corresponds to a packed conduit filled with particles of free space, there is less degree of freedom than in a packed conduit filled with particles which have a solid skeleton. This boundary condition, in turn, results from the Laws of Nature which dictate that solid matter and free space are mutually exclusive. It follows, therefore, that the values of λ, d c , δ in equation (13) will be uniquely defined when these four boundary conditions prevail. Thus, setting four boundary conditions in Quinn's Law for the values of δ=1/8, d p =D, n p =-n pq and ε p =1, establishes a unique value for λ at any given flow rate, when the Forchheimer type coefficients a, and b, are known. We emphasize, however, that the technique outlined above of using plots of equations (3) and (4), as shown in Figures 1 and 2, will not produce accurate values for the Forchheimer coefficients a, and b, in the case of an empty conduit, since they are only capable of returning an average value for these coefficients and, of course, the value of b in an empty conduit varies as a function of flow rate.
Thus, Quinn's Law is the only fluid flow model capable of validating the accuracy and precision of the underlying conduit variables of D, L, and k, in an empty conduit, based upon Forchheimer type measurements of fluid flux and hydraulic conductivity.
We digress, once again, to explain the significance of Quinn's Law as it pertains to the independent variable, k, the roughness of the inner conduit wall in an empty conduit. There are just three measureable variables for an empty conduit, D, L, and k. The former two variables are easy to measure and, in addition, can usually be measured with a high degree of accuracy. This is not the case with the latter variable, k which is very difficult to measure, in the first instance, not to mention the accuracy of the measurement. Accordingly, in the case of an empty conduit, one can use the Forchheimer equivalent type measurements for an empty conduit, in conjunction with Quinn's Law, to accurately back-calculate for the value of k, when the measured data contains pressure drop measurements taken at sufficiently high values of the modified Reynolds number, where the secondary wall effect, i.e., the wall roughness coefficient, manifests itself by punching through the ever-dissipating boundary layer. This is a very effective tool for the practitioner.

Illustrative Examples
We shall now present illustrative examples for all of the categories above as well as an example for roughened walls.
Our first illustrative example is based upon our own laboratory experiment involving a conduit of dimensions 0.3681 x 97.75 cm into which we packed 297 stainless steel ball bearings with a value of d p =0.3291 cm, to which we have given the identification label of HMQ-14. [49]. In our Figure 3 below, we have included the results of our procedure for determining the values of the underlying packed conduit variables, in this case a packed conduit with a low value of D/d p =1.12.  . Packed conduit with low D/dp ratio data plotted after equations (3) and (4).
As can be seen in Figure 4, neither of the plotted graphs generate the correct Forchheimer coefficient values for a, and b, and, in addition, each plot generates a constant value for each coefficient but with slightly different magnitudes.
Finally, in our Figure 5 below, based upon the teaching of Quinn's Law, we show a complete picture of the performance characteristics of this packed conduit over the corresponding flow rate range of 55 to 385 mL/min. We emphasize here that in the plot of dimensionless permeability in Figure 5, the measured values appear to overlay the line for λ=1. This is because we typically show this plot as a log-log plot. If, on the other hand, we simply focused on the narrow range of Q N values measured in this example, and used linear coordinates, we would see that the measured values are offset slightly from the line for λ=1. We offer this as a cautionary note to the reader because in many instances in the published literature, and especially in this field of study relative to the scaling of experiments over a very broad range of modified Reynolds number values, unscrupulous authors have used log-log plots to hide empirical evidence which does not otherwise support their theoretical narratives.
Our second illustrative example is taken from a paper by Sidiropouloe, et al., published in 2007 [50]. In that paper, the authors present empirical data for 115 different packed conduits with many different types of particles.
It is noteworthy to point out, initially, that in samples numbered 1 through 26 in the paper, there are no values provided for the Forchheimer coefficient b. Accordingly, it is apparent that for these samples, the authors totally ignored kinetic contributions and, consequently, as pointed out above, their reported values for the underlying packed conduit variables cannot be validated based upon this empirical evidence.
In our Figure 6 below, we have selected some of the samples from their paper in which the authors reported both Forchheimer coefficient values for a, and b. In addition, we note that in our subset of reported samples, there is only a constant value reported for both coefficients. Accordingly, we assume that the ratio of D/d p was large in all cases, a critical experimental detail which we could not find emphasized anywhere in the paper. As shown in our Figure 6, we have included the results of our procedure for determining the values of the underlying packed conduit variables. Furthermore, we have included a comparison between our calculated values for the external porosity, ε 0 , and the values reported by the authors. As shown in the figure, there are some very significant differences for the values of this variable which we attribute to the experimental methodology used by the authors to measure the value of the external porosity, ε 0 , and/or the particle diameter equivalent, d p , neither of which, we contend, was sufficiently accurate or precise. These discrepancies are symptomatic of the typical experimental protocols used in the engineering disciplines regarding fluid flow through packed conduits.  Example 3.
Our third illustrative example is based upon our own laboratory experiment involving a peak capillary tubing of nominal dimensions 0.02 inches in diameter and 12 inches in length [51]. In our Figure 8 below, we have included the results of our procedure for determining the values of the underlying packed conduit variables, in this case an empty capillary. In addition, in our Figure 9 below, using the teaching of Quinn's Law, we show a complete picture of the performance characteristics of this empty conduit over a flow rate range of 20 to 125 mL/min. As shown in Figure 8, the value of the Q-modified Ergun coefficient A, and the value of the Forchheimer coefficient a, are constant at all Reynolds number values, but the values of B, b and λ change as a function of the modified Reynolds number R em . Since this capillary has a smooth inner wall, the changing values of the three parameters mentioned, i.e., B, b and λ, are due to the primary wall effect whose impact on permeability is most pronounced at modest Reynolds number values. Note that the value of λ in this scenario is relatively large, since an empty conduit has a low and constant value for the τ parameter.   It is obvious from Figure 9, in the plot of Θ v Q N , that in this example, the primary wall effect has a negligible impact at very low values of Q N , i.e., it coincides with the line for λ=1, but manifests significantly through moderate values of Q N and, if projected in a straight line in our mind's eye, would coincide once again with the line for λ=1, at very high values of Q N . This is a manifestation of the dissipation of the boundary layer as the fluid becomes fully turbulent at high values of Q N .
Our fourth illustrative example is based upon the work reported by Nikuradze [38]. This is the classical study carried out circa 1930 which involved the grafting of particles of sand onto the inner surface of drawn brass pipes to create six levels of wall roughness criteria. Even to this day, this study is considered the gold standard of wall roughness empirical data. We have plotted Nikuradze's measured data according to the teaching of Quinn's Law for dimensionless permeability in our Figure 10 below. As shown in Figure 10, we have superimposed Nikuradze's measured results on the underlying background of the λ parameter taught in Quinn's law, in our plot of dimensionless permeability. As shown in the plot, Nikuradze's roughened data falls within a range of λ values from 2 to 11, approximately. It is obvious from the plot that the secondary wall effect, i.e., the roughness coefficient, manifests at sufficiently high modified Reynolds number values as the boundary layer is progressively dissipated, commensurate with the magnitude of the roughness coefficient of a particular conduit under study.

Evaluating Third Party Published Works
We shall now evaluate a cross-section of published articles in the literature, applying our solution to the N-S equation using the teachings of Quinn's Law, to pinpoint discrepancies which are commonplace in the literature due to inadequate measurement techniques. To highlight the discrepancies, our Equation for Fluid Flow in Closed Conduits analysis will generate the Q-Modified Ergun constants, A, and B, the Forchheimer coefficients, a, and b, the value of λ  as well as our back-calculated values for the conduit variables, for each sample evaluated. For guidance in our analysis, we shall follow a precise accept/reject procedure as follows; We shall apply the Q-modified Ergun equation as an accept/reject criteria for any reported packed conduit data having solid particles. In particular, we shall use the value for the B coefficient as our measurement standard with respect to conduit packed density, i.e., external porosity. Therefore, consistent with our own experience of packing many conduits with a variety of rigid particles (the packed conduits being numbered in the hundreds of thousands over a career of 30 plus years), which dictates that well-packed conduits have a range of external porosities of 0.36 to 0.47, and based upon the teaching of Quinn's Law for the value of the coefficient B, our accept/reject criteria for the value of B is: 1.5 < B < 3.5. The value of B=1.5 represents the highest value of external porosity possible, i.e. ε 0 =0.47, and a value of B=3.5 represents the lowest value of external porosity possible, i.e., ε 0 =0.36.

Example 5: Data of Banerjee et al 2017 & 2019
In a recent published article (2019), whose title would seem to suggest that the authors had discovered something important related to permeability "Over the Complete Flow Regime", Banerjee et al claim that they have made a correlation between the Forchheimer coefficients a, and b, which dictates that the values of the coefficients are a function of the fluid flow profile [52]. In fact, they define "corrections" to several parameters in their 2017 paper and "create" a parameter which they designate as, γ, whose value changes over the fluid flow regime and suggests that both values of the Forchheimer coefficients a, and b, are a function of this γ parameter. We categorically disagree with the author's conclusions, as well as the "corrections" upon which they are based.
Firstly, the authors make it a point to present hydraulic conductivity data representing external porosity values of packed conduits ranging from 0.30 to 0.50. This range is far too broad based upon our experience of packing many conduits with various rigid particle types and, in addition, their paper lacks any meaningful analysis of the role of external porosity or the care with which they measured its value, which variable is clearly the most sensitive element impacting their reported development.
To illustrate our arguments, we have applied our N-S solution for closed conduits to a subset of the data reported in this paper and captured our analysis results in Figure 11 below. As can be seen in Figure 11, we have assumed a D/d p ratio of 10, for all the packed conduits in our sample set, thus eliminating all wall effects. The reported hydraulic conductivity data in this paper renders values for the Qmodified Ergun constant B which is consistent with our accept/reject criteria. It is apparent, however, that the author's values for external porosity are significantly larger (10-20%) than those generated by our application of the N-S equation using Quinn's Law. This discrepancy is at the heart of the author's methodology because, as dictated by Quinn's Law, both the Forchheimer coefficients a, and b, are very sensitive functions of this parameter as can be seen from our equations (5) and (6) above. Moreover, the coefficient b is an extremely sensitive function of the value of the external porosity, which means that any discrepancy in porosity values will produce greatly exaggerated errors in predicted values for hydraulic conductivity when kinetic contributions dominate.
In our Figure 12 below, we show the relationship dictated by Quinn's law between the values of the Forchheimer coefficients a, and b, and external porosity, for a conduit packed with 5mm spherical particles at a D/d p ratio of 10. In this scenario, our λ value is 1.00. In Figure 13 below, we show the teaching of Quinn's law relative to hydraulic conductivity and the Forchheimer coefficients over a wide range of modified Reynolds number, for a conduit packed with 5mm spherical particles, at an external porosity of 0.30 (suggested by the paper's authors) and a D/d p ratio of 10. As can be seen from Figure 13, the values of the Forchheimer coefficients a, and b, remain constant over an extended range of modified Reynolds number values, R em , i.e. 2-3,500, which represents a significantly changing fluid profile. This conclusion is at odds with the assertions made by the authors in their paper.
Finally, it is noteworthy to point out that the paper author's development is based upon the manifestly absurd assumption that "The flow path is assumed to be straight and undeviating through the porous packing". One wonders how the authors could rationalize this false assumption with their statement that they used empirical data to validate their theoretical development, since everyone knows that the flow is decidedly sinuous in conduits packed with crushed stones.

Example 6. Data of Shrinivas et al 2014
In this worked example, the authors reported on the permeability of various crushed rock [53]. The paper includes results for the "present study" as well as data by Nasser and Niranjan. In our Figure 14 below, we have captured our analysis for this data set. As can be seen in Figure 14, a significant number of the reported samples fall outside our accept/reject criteria for external porosity (highlighted in bold). In addition, the author's reported values for d p , the spherical particle diameter equivalent, and ε 0 , the external porosity, are significantly different from our analysis values, and are indicated in the columns labeled ∆d p % and ∆ε 0 %, respectively. The discrepancies are significant and vary in both directions, i.e., some are larger and some are smaller than our values. We suggest that the experimental protocols used in this paper were not sufficiently accurate or precise to support the author's conclusions.  As can be seen in Figure 15, again two of the conduits fall outside our acceptable window with respect to external porosity. In addition, it is also clear that the author's experimental protocol for determining the values of d p and ε 0 were not adequate. Their reported values for both these variables vary significantly in both directions as compared to our N-S solution for both rounded and crushed rocks.

Example 8. Data of Zhongxia Li et al 2019
In our Figure 16 below, we present our analysis of the data set reported in a paper by Li et al. published in 2019 [55]. As can be seen in Figure 16, just 4 of the samples fall outside our acceptable window for external porosity. In addition, it is also clear, again that the author's experimental protocol for determining the values of d p and ε 0 were not adequate. Their reported values for both these variables vary significantly in both directions as compared to our N-S solution for these particles of quartz sands. Furthermore, since the authors state that all packed conduits had the identical same value for external porosity, i.e., 0.36, we are inclined to reject the entire premise of this paper as an example of empirical data, but include it as part of our selected examples to alert the reader to the fact that the literature is filled with these type of "measurement" examples.

Example 9. Data of Neue et al 2005
In our Figure 17 below, we include our evaluation of a paper by Neue et al., published in 2005 [56]. This paper represents a category of packed conduits of commercial value for the high efficiency analysis of small molecules of interest in the pharmaceutical industry. The chromatographic "columns" as they are referred to in the industry, are used in the application of HPLC, i.e., high pressure liquid chromatography. As can be seen in Figure 17, these packed conduits contain small spherical particles operated at relatively high pressure drops. Note that there is relatively good agreement in our analysis between the values reported by the paper authors for particle diameter. However, the values for external porosity reported by the authors are typically 10-15% too low, Equation for Fluid Flow in Closed Conduits representing as they do: (a) an embedded assumption in the author's paper, that the value of the constant A in the Kozeny-Carman equation is always 180, and (b) the fact that only a solitary measurement of permeability was made for each sample at extremely low values of the modified Reynolds number, thus, totally ignoring kinetic considerations.

Example 10. Data of Cabooter et al 2008
In our Figure 18 below, we include our evaluation of a paper by Cabooter et al. published in (2008) [57]. This paper is also an example of HPLC columns but represents the most modern version thereof, typically referred to in the industry as UHPLC, i.e. ultra high pressure liquid chromatography. Additionally, this paper allows us to underscore a fundamental issue which appears time and time again, particularly, in the chromatographic literature. That issue is the fact that the Laws of Continuity are generally ignored with respect to the utilization of free space within a packed conduit. This issue raises its' head in this example, since the authors report the use of several different particle diameter choices but report just one set of external porosities and one set of pressure drop measurements and, moreover, totally ignore the number of particles, or an equivalent reconciliation, for each of the respective sets of particle diameter values reported. We have chosen herein just one of the reported sets of values for particle diameter, i.e., that provided by the manufacturer of the particles. As shown in Figure 18, the particle diameters are even smaller in these type columns and are typically in the range of 2-3 microns. Again, note that there is relatively good agreement in our analysis between the values reported by the paper authors for particle diameter. However, the values for external porosity reported by the authors are, once again, typically 10-20% too low, due to: (a) an embedded assumption in the author's paper, that the value of the constant A in the Kozeny-Carman equation is not constant, but varies as a function of particle size distribution and, (b) the authors totally ignored the porosity of these particles, ε p , (an independent variable whose value is available from the particle manufacturer) and, accordingly, made no attempt to reconcile the packed conduit internal porosity, ε i , thus, violating the Laws of Continuity for packed conduits and, (c) the fact that only a solitary measurement of permeability was made for each sample, at extremely low values of the modified Reynolds number, thus, totally ignoring kinetic considerations.

Conclusions
In this paper, we have demonstrated a unique solution to the Navier-Stokes equation for closed conduits, both packed and empty. In so doing, we have used both the underlying theory and equation expressed in Quinn's Law, to arrive at a boundary condition which identifies the value of λ at one specific flow rate for any given fluid flow experiment under study. This is both a necessary and sufficient condition to identify, simultaneously, the unique combination of the values of d p and ε 0 in any closed conduit under study, and which, in combination with the measured Forchheimer coefficients of a, and b, can correlate precisely any measured data at any point across the entire spectrum of the modified Reynolds number, which includes the regions typically referred to in conventional wisdom as, laminar, transitional and fully turbulent. We have also evaluated several examples of empirical data including both packed and empty conduits, from our own laboratory as well as those taken from published third party laboratories. Our conclusions based upon our application of this unique solution to the N-S equation can be catalogued as follows: 1. The value of the coefficient A in the re-invented Ergun model, i.e., the Q-modified Ergun equation, has the constant value of 256π/3 which is 268.19 approx. 2. The value of the coefficient B in the re-invented Ergun model, i.e., the Q-modified Ergun equation, is a variable function of the external porosity of the conduit as well as the wall normalization coefficient λ. Thus, the typical value of B falls in a range from 1.5 to 3.5 for wellpacked conduits packed with rigid particles having a solid skeleton and a large ratio for D/d p . 3. There is but one unique combination of values for d p , ε 0 and n p which will correlate permeability measurements over the entire range of the fluid flow profile, from creeping flow to fully turbulent flow. This is a result of the Laws of Nature which dictate that for any given conduit under study, every combination of the packed conduit values of the variables d p and n p represents a unique hypothetical Q channel and, consequently, a unique pressure gradient/fluid flow rate profile over the entire fluid flow regime. This relationship between pressure gradient and fluid flow rate is quadratic over the entire range of flow rates in any given packed conduit under study. 4. The Laws of Continuity dictate that for any given conduit dimensions, i.e. diameter D and length L, packed with rigid particles of spherical particle diameter equivalent, d p , the value of ε 0 is not an independent variable but is defined by the combination of the values of d p and n p , i.e. the number of particles present in the packed conduit. 5. In order to accurately define the Forchheimer coefficients a, and b, empirical measurements must be taken in the nonlinear portion of the fluid flow regime, i.e. where kinetic contributions are significant. This is a result of the fact that kinetic contributions are much more sensitive to the value of the external porosity parameter, ε 0 , than are viscous contributions. 6. In most engineering examples studied in the literature, the experimental protocols typically used to measure the values of d p and ε 0 are not sufficiently accurate or precise to validate the value of these variables using pressure drop measurements. Measurement discrepancies occur in both directions, i.e., both larger and smaller than the actual values. Thus, the values reported in the literature have values for external porosity which are both too high and too low. 7. In most chromatographic examples studied in the literature, on the other hand, the experimental protocols typically used to identify the values of d p and ε 0 are fundamentally flawed for several reasons. Firstly, they are based upon a back calculation of permeability which is derived based upon the erroneous assumption that the value of the coefficient in the Kozeny-Carman fluid flow model is either constant at 180 approx., or varies as a function of the particle size distribution, secondly, the values for d p and ε 0 are both independently measured, without any reconciliation with the value of n p , thus violating the Conservation Laws as applied to the total amount of free space contained within the empty conduit under study, thirdly, pressure drop measurements are typically taken in the linear portion of the fluid flow regime (laminar) thus totally ignoring kinetic contributions, fourthly, particle porosity is sometimes substituted for conduit internal porosity and, finally, mobile phase velocity is sometimes substituted for superficial fluid velocity. These errors typically result in reported values of permeability, which do not facilitate accurate identification of underlying fluid embodiment variables based upon measured pressure drops, most notably external porosity, which is invariable 10-15% too low.

Declaration of Interest
The author declares no conflict of interest.