Page 1 of 23
European Journal of Applied Sciences – Vol. 11, No. 2
Publication Date: April 25, 2023
DOI:10.14738/aivp.112.14344.
Quinn, H. M. (2023). A Fluid Dynamic Development Like None Other. European Journal of Applied Sciences, Vol - 11(2). 407-429.
Services for Science and Education – United Kingdom
A Fluid Dynamic Development Like None Other
Hubert M Quinn
Department of Research and Development,
The Wrangler Group LLC, 40 Nottinghill Road, Brighton, Ma.02135, USA.
ABSTRACT
For more than 100 years, much uncertainty has surrounded fluid flow
characteristics, both in open pipes and packed conduits. Not the least amongst this
uncertainty has been the physics of the viscous boundary layer, in the former, and
the so-called “wall-effect”, in the latter. Indeed, no fluid flow model has ever been
able to reconcile both packed and empty conduits within the same theoretical
framework, until now. In this paper, we describe the Quinn Fluid Flow Model
(QFFM), which was first published in 2019 and represents the first model to
capture, seamlessly, the physics of both empty and packed conduits, including all
elements of wall-effects. This is not the only distinguishing feature of the model,
which also includes, again, for the first time, a theoretical description of the so- called “constants” within the pressure flow relationship in closed conduits.
Consequently, this model enables fluid dynamic validation over the entire fluid flow
regime from creeping flow, at low Reynolds number values, to fully turbulent, at
high Reynolds number values. This unique feature is possible because, on the one
hand, in empty conduits, extremely large Reynolds number values are achievable at
reasonable pressure drops but, very low Reynolds number values are not, due to
the pressure drops being prohibitively low. In packed conduits, on the other hand,
very low Reynolds number values are achievable at reasonable pressure drops but,
large Reynolds number values are not, due to the pressure drops being
prohibitively high. Accordingly, the QFFM enables simultaneous validation, at high
Reynolds number values in the former and, at low Reynolds number values in the
latter. This characteristic, therefore, to validate over the entire fluid flow regime
using the same underlying physical and mathematical framework, sets this fluid
flow model apart from all others. In addition to highlighting the many features of
the QFFM in this paper, we will also demonstrate a comprehensive validation over
ten orders of magnitude of the Reynolds number using published classical studies,
as well as home grown experiments. To our knowledge, no extant fluid flow model
comes even close to this comprehensive description of the fluid dynamics of fluid
flow, not only in closed conduits, but also, by extrapolation, in other applications
outside of this very narrow field of study.
Keywords: Packed Conduits; Empty Conduits; Laminar; Transitional; Turbulent.
INTRODUCTION
In this introduction, we will deviate slightly from conventional practices wherein a detailed
rehash of well-known literature is usually provided. Instead, we will briefly mention the four
best known fluid flow models, two for packed conduits, two for empty conduits and, in addition,
attach, for the benefit of the practitioner, the actual Matlab program for the QFFM, in a series of
sequential steps designed for the Matlab software. In this manner, a practitioner who is familiar
Page 2 of 23
Services for Science and Education – United Kingdom 408
European Journal of Applied Sciences (EJAS) Vol. 11, Issue 2, April-2023
with the engineering Matlab technology can simply start to use the QFFM and thereby become
familiar with its new teachings, without further ado. We suggest that, in general, the best way
to understand something new is to start to use it for one’s everyday tasks.
In the case of packed conduits, the two best known fluid flow models are the Kozeny/Carman
[1] and Ergun [2] models, the former being valid only in Laminar flow and the latter,
supposedly, for all other forms of the fluid flow regime. Unfortunately, however, neither flow
model can be validated due to severe shortcomings contained in their underpinnings. The
Ergun model is actually an extension of the Kozeny/Carman model wherein a second term was
added to account for pressure drop losses due to kinetic contributions. Both models suffer from
the same ailment, however, in that both are based upon the use of fixed (erroneous)
coefficients, the so-called constants, and neither one can account for wall-effects. Similarly, in
the case of empty conduits, the best-known models are the Poiseuille model [3] and the
Blausius model [5], both of which are unsatisfactory, also, because they cannot be validated
over the entire fluid flow regime, i.e., from creeping flow to fully turbulent. Accordingly, this
field of study is ripe for a model which does exactly that, and hence the need for the QFFM [6].
In our Table 1 herein, the code for the QFFM using Matlab software is disclosed within an excel
spreadsheet. This sequence of code steps can be simply loaded directly into the Matlab software
and used to solve problems immediately. Each step contains a specific description which is self- explanatory and are presented in units of the cgs convention. The first 17 steps of the code
contain a description of the Hypothetical Q Channel (HQC) which represents the unique
theoretical underpinnings of this model. This development is unique amongst all extant models
because it defines a packed conduit, as the general case and an empty conduit, as a special case.
In both cases, however, the HQC is defined based upon the terms of a packed conduit containing
particles which can vary in porosity from zero (ep = 0) at one end of the spectrum, i.e.,
nonporous particles, to particles which have a porosity of unity (ep = 1) at the other end of the
spectrum, i.e., fully porous particles (empty conduit), and particles having all intermediate
values of porosity (0 < ep < 1), i.e., partially porous particles. This theoretical underpinning of
fluid flow in closed conduits is the direct opposite to all extant fluid flow models which equate
a packed conduit to a “bundle of crooked tubes” [6]. Steps 18 through 53 contain the necessary
variables and formulae computations which define any fluid flow embodiment under study.
This is the portion of the code which contains the vital permeability related information. Steps
54 through 74 contain the information regarding fluid flow profile which is uniquely described
in the QFFM as that of a harmonic oscillator. Accordingly, the fluid motion is defined as damped
simple harmonic motion, with the two distinct damping mechanisms being wall friction, on the
one hand, when the flow rate is small (laminar) and fluid friction, on the other hand, when the
flow rate is large (turbulent). Steps 75 through 82 contain what we refer to herein as the “QFFM
solution for permeability” in the fluid dynamics of closed conduits. It provides for the
practitioner a simple methodology of using permeability measurements, i.e., flow rate and
pressure drop measured data, to back-calculate for the fundamental conduit parameter of dc,
the diameter of the HQC representing the flow embodiment under study. This back-calculation
establishes the two dependent variables of dp, average spherical particle diameter equivalent
and particle fraction, fp, by isolating the independent variable of np, the number of particles of
diameter dp contained within the flow embodiment under study. Finally, steps 83 through 99,
allows the practitioner to generate a full set of performance graphs showing the complete
Page 3 of 23
409
Quinn, H. M. (2023). A Fluid Dynamic Development Like None Other. European Journal of Applied Sciences, Vol - 11(2). 407-429.
URL: http://dx.doi.org/10.14738/aivp.112.14344.
performance characteristics of any fluid flow embodiment under study. As an illustration of
what the QFFM can deliver via the Matlab software, we include Fig 1 and Fig 2 herein.
Table 1
Page 4 of 23
Services for Science and Education – United Kingdom 410
European Journal of Applied Sciences (EJAS) Vol. 11, Issue 2, April-2023
Table 1 Continued
Page 5 of 23
411
Quinn, H. M. (2023). A Fluid Dynamic Development Like None Other. European Journal of Applied Sciences, Vol - 11(2). 407-429.
URL: http://dx.doi.org/10.14738/aivp.112.14344.
Fig. 1 (steps 95 through 99)
Fig. 2 (Steps 93 through 94)
THE QFFM - A NOVEL CONCEPT IN FLUID DYNAMICS
The Hypothetical Q Channel (HQC)
We begin this topic in our development by establishing the pivotal role played by the actual
number of particles, np, having a diameter of dp, which are theoretically present in any given
conduit under study, whose volume is defined by its diameter, D, and length, L. When we define
Page 6 of 23
Services for Science and Education – United Kingdom 412
European Journal of Applied Sciences (EJAS) Vol. 11, Issue 2, April-2023
particles in this development, we define them based upon three independent characteristics,
namely, the measured particle diameter, dpm, the particle sphericity, Wp, and the measured
particle porosity, ep. The HQC, on the other hand, is defined by its characteristic dimension
(diameter) dc = dp/abs(np/npq) and is disclosed as step #17 in Table 1 herein. Note that dp, is a
packed conduit characteristic, not an actual particle characteristic, and is , therefore, not an
independent variable, but rather a dependent variable in the pressure/flow relationship (step
# 10). The particle sphericity term, Wp, is considered a “fudge factor” since no one has yet
published a means to measure it accurately, especially in the case of non-spherical particles.
We include herein, Fig 3 and Fig 4 to illustrate the relationship between the values of dc and np.
Fig 3
As shown in Fig 3, for a conduit packed with particles where 0 ≤ ep ≤ 1, an overlay of the plot of
the conduit porosity function, e versus np, and the plot of dc versus np, shows how the conduit
particle fraction (np/npq) and the conduit external porosity (1-np/npq) vary as a function of the
value of np, the number of particles present in the packed conduit. This plot also shows that
when the value of np = 0, the conduit porosity function is discontinuous because the value of dc
is infinite, which means that the value of external porosity can never be unity in any packed
conduit under study (a novel concept not found in conventional wisdom). Note that in an empty
conduit, i.e., when the value of np = -npq, the value of the conduit external porosity(1-np/npq) is
2, another novel concept not found in conventional wisdom, but one that is a consequence of
simple algebra in the QFFM, i.e., 1-(-1) = 2. We emphasize, also, that when the particles contain
a solid skeleton, which means they are either nonporous or partially porous, the value of np
must, by necessity, be less than the value of npq (the Kepler Conjecture). This is in contrast to an
empty conduit, which means the particles are fully porous, i.e., they are comprised of free space,
in which case, the value of np is always equal to the value of npq (another novel concept not
found in conventional wisdom). Note also that the absolute value of the particle fraction is used
to define the value of dc, thus, ensuring that it is always positive which is a necessary boundary
condition for its value.
Page 7 of 23
413
Quinn, H. M. (2023). A Fluid Dynamic Development Like None Other. European Journal of Applied Sciences, Vol - 11(2). 407-429.
URL: http://dx.doi.org/10.14738/aivp.112.14344.
Fig 4
As shown in Fig 4, the conduit internal porosity is negative in a packed conduit containing fully
porous particles, i.e., an empty conduit (another novel concept not found in conventional
wisdom). Therefore, considering the theoretical underpinnings of a packed conduit in the
QFFM, one can clearly differentiate this development from any other in the realm of fluid
dynamics.
Note of Clarification: Since we have emphasized herein the pivotal role of the term np, the
number of particles present in any packed conduit under study, we want to underscore some
practical elements of how a practitioner would actually count the number of particles in a
validation experiment. In order to maintain a reasonable number of particles to be counted and,
simultaneously, the diameter of which one can measure accurately, both prerequisites for a
validation experiment, the packed conduit must, by definition, have a small conduit-to-particle
diameter ratio, i.e., D/dp. In addition, all the particles need to be perfectly spherical, and have
the same dimensions, such that the terms dpm and dp are identical, i.e., Wp = 1. This is best
achieved in an empty conduit when, Wp = 1.0 and dpm = dp = D. This means, of course, that the
wall-effect will be significant which is not an issue with the QFFM, as explained below, but
would, of course, not be possible with other models which cannot accommodate the wall-effect.
In packed conduits where it is not possible to actually count the number of particles, i.e., at very
large D/dp ratios or when the particles are not perfectly spherical and/or are very small in
diameter, other means can be used as an alternative, such as measuring the mass of particles
packed into the conduit. However, since these techniques all have additional uncertainties
associated with them, we recommend that the QFFM solution for permeability (steps 75 to 82
Page 8 of 23
Services for Science and Education – United Kingdom 414
European Journal of Applied Sciences (EJAS) Vol. 11, Issue 2, April-2023
in Table 1) should be used to back-calculate the value of np, as discussed below, using
permeability measurements which are generally achieved with very low experimental
uncertainty. This recommendation is driven by the reality that the conduit external porosity,
i.e., (1-pf) cannot be measured accurately/independently, especially when the particles are
partially porous, and is, therefore, by definition, a residual calculation, i.e., the remaining space
within the packed conduit after the particle fraction has been accounted for, as dictated by the
Laws of Nature. Thus, we are saying that this back-calculation technique trumps any direct
measurement technique for conduit external porosity.
The Pressure/Flow Relationship based upon the HQC
We begin this topic by introducing what has now become known as Quinn’s Law of Fluid
Dynamics.
PQ = k1 + k2CQ (1)
Equation (1) is disclosed as step # 50 in our Table 1 herein and is the dimensionless
manifestation of Quinn’s Law. Although this equation is about fluid dynamics, you would not
know it by just looking at it, because it simply represents a straight line. In the case of equation
(1), on a plot of PQ versus CQ, k1 represents the intercept on the y axis and, and k2 represents
the slope of the line as shown in Fig. 5 for pressure/flow measurements taken by this author in
a Teflon capillary (empty conduit) of 3 different lengths, 15,240 cm, 7,620 cm and 241 cm.
Fig. 5
So, what ties equation (1) to the study of fluid flow in closed conduits, as opposed to being tied
to some other relationship found in Nature or simply some unknown random phenomenon?
The answer to this question is that all the knowledge is hidden in the meaning of the 4 terms in
equation (1). So let us unwrap it step-by-step.
Firstly, equation (1) is a dimensionless relationship which means that each of the four terms
which make up the relationship has embedded within them a number of variables found in
Page 9 of 23
415
Quinn, H. M. (2023). A Fluid Dynamic Development Like None Other. European Journal of Applied Sciences, Vol - 11(2). 407-429.
URL: http://dx.doi.org/10.14738/aivp.112.14344.
Nature, the dimensions of which, when combined, cancel each other out. So, one can think of
equation (1) as being a relationship where all the variables have been normalized out of the
relationship and, thus, it stands as a universal relationship, which means that no matter what
kind of experiment one runs to evaluate this relationship in Nature, the results of the
experiment will always fall on this straight line. This is an amazing statement, if true. So, to
prove it, let us begin at the beginning.
Equation (1), in mathematical jargon, tells us that when a fluid is forced to flow through a closed
conduit, i.e., pipe, the entity represented by PQ found in Nature, is the sum of two distinct
processes. Those processes we shall call viscous contributions and kinetic contributions. Thus,
in equation (1) we can identify the terms as follows;
PQ = k1 + k2CQ
(Viscous type friction factor) = (Normalized Viscous contributions) + (Kinetic contributions).
Therefore, equation (1) can be referred to as a viscous type friction factor, because we have,
advantageously, normalized the relationship for viscous contributions. We can see this
immediately by looking at the equation, since the viscous term is comprised of only a constant
value, k1. This results from our normalization process referred to above, where we have divided
the equation across by all the embedded variables in the viscous term. So, already, we can see
that what looked like an innocent straight-line relationship is anything but. So, a reasonable
question to ask is, where does these four terms come from in Nature?
Viscous contributions arise when fluid passes by a solid object and the interaction between the
flowing fluid and the solid surface creates an interference which produces a retarding force on
the motion of the fluid. Accordingly, it would be logical to suggest that this retarding force
would be proportional to the surface area of the solid object in the path of the fluid. Thus, one
would expect a larger object, which has a higher surface area, to create a larger retarding force.
Conversely, kinetic contributions arise due to the actual motion of the fluid itself. So, when we
are talking about fluid motion through a closed conduit, it would be intuitive to think that the
larger the opening of the conduit, i.e., its diameter value, the smaller would be the resistance to
the fluid motion. Thus, by just using our intuitive knowledge of the world around us, we could
suggest that in the case of viscous contributions, bigger gets us more resistance to fluid flow,
whereas, in the case of kinetic contributions, bigger gets us less resistance to fluid flow.
Accordingly, we now postulate that the one mechanism is the reciprocal of the other, in terms
of how the Laws of Nature regulate fluid flow in closed conduits.
Already, we have made great progress in understanding fluid flow and we have not yet moved
away from our dimensionless equation (1). So let us now take the next step in how Nature
works.
Consider, for simplicity’s sake an element of free space in the universe, which is a perfect
sphere. We choose a perfect sphere because the Greeks, many moons ago, figured out how to
characterize a sphere in terms of its diameter, D, surface area, D
2
, cross sectional area, D
2
/4,
and volume, D
3
/6, all in terms of a universal constant which has become commonly known as
Page 10 of 23
Services for Science and Education – United Kingdom 416
European Journal of Applied Sciences (EJAS) Vol. 11, Issue 2, April-2023
We accept, herein, from the Greeks, that is a universal constant and has a value that
approximates to the vulgar fraction of 22/7, i.e., close to the value of 3.14. The Laws of Nature
dictate that whenever anything occurs in Nature, all matter must be conserved. These Laws are
referred to as the Conservation Laws. This means that when we involve the free space found in
Nature in any of our experiments, we must account for it all, meticulously. So, let us apply the
conservation Laws to our spherical element of free space and see what we can discern.
Firstly, assume that our spherical element of free space is taken up by a perfectly smooth
spherical particle, say, a glass bead or an electro-polished stainless steel ball bearing, the
diameter of which we now designate as dp. How would we interpret the Laws of Nature with
respect to the two processes at work in fluid flow, i.e., viscous and kinetic contributions,
involving this spherical particle? We know that the surface area of the spherical particle has the
formula dp
2 and so we know that the viscous contributions will be related to that expression
of our free space. On the other hand, if we wanted to represent the kinetic contributions of our
spherical element of free space, we know that the free space would have to flow through an
orifice whose diameter is related to the cross-sectional area, dp
2
/4, of that free space. Thus, we
now have come to the conclusion that the common denominator between viscous and kinetic
contributions is the ratio of the surface area to the cross-sectional area of our spherical particle,
i.e., (dp2
) / (dp2
/4) = 4. Let us now define the term rh = 4 where, rh represents the normalization
factor between viscous and kinetic contributions. In other words, since in equation (1) we have
the sum of these two processes, we must be able to add them together in comparable units, so
rh becomes our “exchange rate” between viscous and kinetics contributions, much as we need
an exchange rate to add together dollars and pounds sterling. We call it the exchange rate
because the computed ratio has a value of 4, not unity. If it were unity, we would not need an
exchange rate, i.e., in our analogy, the dollar would have a value exactly equal to the pound
sterling. Because, however, this ratio has a value of 4, we can now think of it as the
normalization coefficient of drag, since it represents the “imbalance” between the surface area
and cross-sectional area of our spherical particle (free space) and, accordingly, since this also
represents the “imbalance” between viscous and kinetic contributions, as it were, it can be
thought of as a normalization coefficient of “drag” between the two.
THE UNIVERSAL CONSTANTS, K1 AND K2.
Let us now set in place a controlling mechanism for viscous and kinetic contributions in our
equation (1). We do this by choosing, advantageously, as the radius of our unit spherical
particle, the value of rh, representing our control unit. This makes it easy for us to normalize
our equation for both viscous and kinetic sources. Thus, the control volume of free space taken
up in our experiment is (4/3)rh
3 and we need to apportion this volume out between viscous
and kinetic contributions, according to the dictates of the Laws of Nature. We select as our
orifice for flow a unit cylindrical channel whose circumference is 2r, where r = 1, is the radius
of the unit channel. Next, we set up the controlling mechanism between viscous and kinetic
contributions, by making the viscous contributions proportional to the area of our control
element, and the kinetic contributions proportional to the reciprocal of the circumference of
our unit flow channel, normalized via our drag normalization coefficient, i.e., rh. Thus, to get our
universal viscous constant, we must divide our volume element by rh , which changes its
dimensional value to area rather than volume, and gives k1 = (4/3)rh
2 = 64/3 = 64.05 approx.,
Page 11 of 23
417
Quinn, H. M. (2023). A Fluid Dynamic Development Like None Other. European Journal of Applied Sciences, Vol - 11(2). 407-429.
URL: http://dx.doi.org/10.14738/aivp.112.14344.
and to get our universal kinetic constant we must apply the rh that we took from the viscous
side and assign it to our flow orifice, i.e., adjust the radius of the unit channel by the value of rh,
which gives k2 = 1/(2rh) = 1/(8) = 0.04 approx. Thus, we can see now that the viscous constant
represents a direct proportionality with surface area of our unit particle size (rh = the unit),
whereas, conversely, the kinetic constant represents a reciprocal proportionality of our channel
circumference (rh = unit) i.e., these two control elements operate in a reciprocal fashion to
restore the balance between viscous and kinetic contributions, as dictated by the Laws of
Nature.
Accordingly, we can now upgrade our equation (1), which becomes:
PQ = 64 + CQ (2)
3 8
Let us now turn our attention to the CQ term on the right-hand side of equation (2) by peeling
back the layers of normalization which it embodies. Stated another way, let us expose the
embedded variables in CQ.
THE FLUID FLOW TERM CQ
We define the term CQ = QN where both and QN are dependent variables. This is shown in
step number 39 of Table 1. Accordingly, let us explain them each in turn. We shall now change
our nomenclature by referring to our equation (1) as the Quinn Fluid Flow Model (QFFM) to
simplify our descriptions.
Firstly, we will tackle the QN term because it is the term that contains the fluid flow term, i.e., it
contains the fluid flow rate term, q.
The Fluid Current Term QN
We define QN = Rem, where represents a conduit porosity related term, i.e., = 1/(1-np/npq)
3
,
where (1-np/npq) is the external porosity of a conduit under study (the residual fraction within
a packed conduit after the particle fraction (np/npq) is accounted for) and Rem is the modified
Reynolds number. We define Rem = 4qdpf/[(np/npq) where, and f are the viscosity and
density of the fluid used in any experiment under study. Accordingly, in the QFFM
nomenclature, we refer to QN as the fluid “current” because it contains all the physical
characteristics within the packed conduit which influence the fluid flow profile. These physical
characteristics contain elements from the fluid, the particles, and the packed conduit itself.
Thus, one could say that of all the terms contained within the QFFM, the QN term is the most
important. At this point, we should point out that the value of QN in any experiment under study
is critical to understanding the relative contribution of the viscous and kinetic sources we
referred to above. For instance, at very low values of QN, say less than unity, the viscous
contributions are dominant and the kinetic contributions are negligible. Conversely, at very
high values of QN, say greater than unity, the kinetic contributions dominate and the viscous
contributions are negligible.
Page 12 of 23
Services for Science and Education – United Kingdom 418
European Journal of Applied Sciences (EJAS) Vol. 11, Issue 2, April-2023
Let us further define Rem = nk/nv, where nk = the kinetic contribution per unit and nv = the viscous
contributions per unit. Thus, we can now upgrade equation (2) as follows:
PQ = 64+ nk (3)
3 8nv
The Wall Effect Term .
We have already stated that our equation (1) has to do with fluid flow through closed conduits,
which means there has to be a wall involved, i.e., the wall of the conduit confines the fluid and
any obstacles in the path of the fluid, to the free space contained within the conduit. Thus, we
have present in our methodology, the so-called “wall effect”. Accordingly, we can say that within
the QFFM, the term deals with the wall effect.
Fig 6
The QFFM teaches that there are just five elements to the wall effect:
(1), primary wall effect (W1);
(2), secondary wall effect (W2);
(3), residual secondary wall effect (W2R);
(4), net wall effect (WN) and finally,
(5), wall normalization coefficient ().
The data plotted in Fig 6 is all taken from third party published papers. The Super pipe data is
contained in the now classical study from Princeton University [7, 8,9]. The Nikuradze smooth
Page 13 of 23
419
Quinn, H. M. (2023). A Fluid Dynamic Development Like None Other. European Journal of Applied Sciences, Vol - 11(2). 407-429.
URL: http://dx.doi.org/10.14738/aivp.112.14344.
data is to be found in reference [10]. The Nikuradze roughened wall data which contains the 6
levels of roughness is to be found in reference [11]. The Klinker data is to be found in the paper
by Buckwald et al [12].
The wall normalization coefficient = (1 +WN) and enters the pressure flow relationship
through the kinetic term. Accordingly, the parameter has a relatively small effect when the
flowrate is low, i.e., when the kinetic contributions are negligible and the value of QN is very
small, i.e., less than unity, say.
The net wall effect WN = (W1 + W2R) and will only manifest in measured pressure drop values
where kinetic contributions are significant. In other words, the net wall effect has a relatively
small effect in laminar flow.
The Primary Wall Effect (W1)
The primary wall effect has the symbol W1 and is a derivative of two distinct parameters.
Firstly, W1 is a derivative of the dimensionless viscous boundary layer 0 = (k1/k1 +k2QN), where
k1 = (64/3) and k2 = (1/8), and are the universal constants derived above. QN = Rem is the
Quinn number. The symbol Rem stands for the well-known modified Reynolds number and =
1/(1-np/npq)
3
.
Secondly, the formula for the primary wall effect is W1 = (0
(1/3)/). It is, therefore, also a
derivative of the tortuosity factor, = where = (npqD/L) is a structural feature of the flow
embodiment under study (packed or empty conduit), where npq is the volume of the empty
conduit expressed in terms of number of particle equivalents having a diameter of dp, the
average spherical particle diameter equivalent.
W1 Has Its Maximum Value W1 = 5.33 Approx:
There is just one operating condition when the value of the primary wall effect has its maximum
value of 5.33 approx. This condition exists when D/dp has its minimum value of unity [(D/dp) =
1.0], which only occurs in a conduit packed with particles of free space wherein the particles
are fully porous ( p =1), i.e., an empty conduit, and when the value of QN is very small. In this
scenario, the tortuosity coefficient, , is relatively small which results in a thick viscous
boundary layer when the value of QN is very small (Laminar flow). As the value of QN increases,
however, the viscous boundary layer starts to dissipate which reduces its thickness and
eventually approaches a value of zero at very large values of QN (fully developed turbulence).
W1 Manifestation:
The primary wall effect W1 will only manifest when the value of (D/dp) is very small, such as in
an empty conduit, and when kinetic contributions are significant. At very high values of QN, on
the other hand, where the boundary layer is totally depleted, the value of W1 = 0, and thus will
not manifest at all. Additionally, when D/dp is very large, say greater than 30, which only occurs
in a conduit packed with solid particles, the tortuosity coefficient, , is very large and this
reduces the thickness of the boundary layer to an infinitesimally small value, hence the value
for W1= 0 approx., at all values of QN.
Page 14 of 23
Services for Science and Education – United Kingdom 420
European Journal of Applied Sciences (EJAS) Vol. 11, Issue 2, April-2023
As shown in Fig. 6 above, an empty conduit with hydraulically smooth walls will have a constant
value of W1 = WN = 5.33 approx., at low QN values. This represents the steady/stable viscous
boundary layer adjacent to the channel wall. At a value of QN > 1 (approx.), the viscous boundary
layer starts to be dissipated and the value of W1 will decrease until it reaches a value of zero at
very high values of the QN number, i.e., fully developed turbulence. This is shown by the
Nikuradze and Super Pipe smooth wall data, which falls in this decreasing region of the viscous
boundary layer because all the measurements were taken at relatively high values of QN, i.e., QN
> 1.
It will be appreciated that W1 = WN, the net wall effect, for all values of QN in a smooth walled
empty conduit.
The Secondary Wall Effect (W2):
The secondary wall effect has the symbol W2 and is a derivative of the sand-grain roughness, k,
and the diameter of the hypothetical fluid channel dc. It is proportional to the ratio of the two
where W2 = 30 kdc(1/3) and kdc = k/dc where k is the sand grain roughness and dc = dp/[abs(np/npq)]
is the diameter of the hypothetical fluid channel. The constant of proportionality is the value of
30, which is due to end effects. It will be appreciated that dp stands for the average spherical
particle diameter equivalent of the particles in the conduit, either packed with solid particles
( p< 1) or particles of free space ( p= 1), i.e., an empty conduit, where p stands for the particle
porosity.
The Value of W2 is Not Always Constant:
When the particle sand grain roughness value, k, has a value greater than zero, i.e., k > 0, the
value of W2 will be a function of the channel diameter dc. A constant value of k, therefore, will
not always produce the same value of W2. The larger the value of the channel diameter dc, the
smaller is the value of W2 for any given value of the sand grain roughness coefficient k. In other
words, wall roughness has less of an impact at larger diameters of the channel.
The Value of W2 Is Only Apparent at High Values of QN When (D/Dp) Is Very Small:
Because, in an empty conduit, the viscous boundary layer may sometimes be greater in depth
than the sand grain roughness value, k, in which case it will mask the effect of the wall
roughness, W2 will only manifest at high values of QN when the viscous boundary layer has been
reduced and the sand grain roughness punches through it.
The Value of W2 Is Always Apparent When (D/Dp) Is Very Large and QN Is Very Large
In packed conduits where the value of (D/dp) is very large, i.e., a typical packed conduit with
solid particles, the secondary wall effect, W2, manifests at all high values of QN wherein the
kinetic contributions are significant.
As shown in Fig. 6, all 10 Klinker packed conduits in our study here with solid particles, have a
range of value for W2 = WN between 1.0 and 6.0 approx. When compared to the Super Pipe data
and the Nikuradze smooth walled data, all the Klinker data fall under the viscous boundary
layer line for smooth walls. This is because, due to the very high tortuosity of the fluid flow in
the Klinker conduits, there is no boundary layer and hence W1 = 0, i.e., the primary wall effect
is negligible.
Page 15 of 23
421
Quinn, H. M. (2023). A Fluid Dynamic Development Like None Other. European Journal of Applied Sciences, Vol - 11(2). 407-429.
URL: http://dx.doi.org/10.14738/aivp.112.14344.
The Residual Secondary Wall Effect (W2R):
The residual secondary wall effect is a derivative of both the primary and secondary wall
effects. Indeed, it is a derivative of the difference between the two, i.e., W2R = W2-W1
(1.2)
.
The Net Wall Effect (WN):
The net wall effect is the sum of the primary wall effect and the residual secondary wall effect,
i.e., WN = W1 +W2R. Itis the net wall effect WN that influences the measured pressure drop.
The only examples in our plot in Fig.6 which have both a primary and secondary wall effect are
the six levels of sand-roughened empty conduits from Nikuradze (levels 1-6 in legend). Note
that all 6 of the Nikuradze roughened conduits fall on the opposite side of the boundary layer
line, represented by the smooth-walled data, as compared to the Klinker data. This is because
although some of the Nikuradze measurements were taken in the region where the boundary
layer masked the secondary wall effect, i.e., the sand grain roughness was buried in the
boundary layer, most of the Nikuradze measurements were taken at high enough values of QN
where the sand grain roughness punched through the boundary layer, hence the appearance of
the line to the right of the boundary layer line. This shows how the residual secondary wall
effect, W2R, manifests in the overall value of .
The Wall Normalization Coefficient ():
We define the wall normalization coefficient = (1 +WN). This is the parameter that accounts
for the impact of wall effect in fluid dynamics. When there is no wall effect, i.e., WN = 0, the value
of = 1.0.
Fig. 7
As can be seen from Fig. 7, a plot of versus QN normalizes for all samples, and allows us to
view their fluid dynamic profile in one single frame of reference. All the Klinker packed
conduits fall between the line for = 1 (no wall effect) and the line for empty conduits with
smooth walls which represents the viscous boundary layer. This is because the Klinker samples
Page 16 of 23
Services for Science and Education – United Kingdom 422
European Journal of Applied Sciences (EJAS) Vol. 11, Issue 2, April-2023
have no primary wall effect and their sand grain roughness equivalent corresponds to a range
of values of less than 6.0 approx.
All the six levels of the roughened Nikuradze conduits falls on the right-hand side of the
boundary layer line. This is because the roughened samples have a residual secondary wall
effect W2R due to the protrusion of the sand-grain roughness through the viscous boundary
layer.
Finally, as can be seen from this analysis, the QFFM is unique amongst all fluid dynamic models,
since it seamlessly accommodates both a packed and an empty conduit, which we will explain
in more detail below. In addition, it provides an exact correlation for empirical data over the
entire flow range, be it laminar through fully turbulent.
THE SOURCE OF THE FLUID DRIVING FORCE PQ
We now focus on the remaining term. PQ, which is the driving force term in the relationship and
is shown in step number 50 in Table 1. We have already described this term as a viscous type
friction factor. So let us now undo the normalization process for the viscous variables and
restore the dimensional variables. We define PQ = P(rhnv ), where P represents the
differential pressure drop across the conduit.
Thus, let us now upgrade our equation (3), by substituting for PQ, as follows;
P = 64+ nk (4)
rhnv 3 8nv
Equation (4) now identifies all the elements embedded in the QFFM. Let us now convert the
elements to the variables which they represent in Nature, and, accordingly, restore the
relationship to its dimensional manifestation in the QFFM. This we will accomplish by simple
straightforward algebra.
Multiply equation (4) across by the normalized viscous contributions, i.e., rhnv. This gives:
P = 256nv+ nk (5)
3 2
Let us define:
(1)the viscous contributions, nv =[(np/npq)
2sL]/((1-np/npq)
3dp
2), and,
(2)the kinetic contributions, nk = [(np/npq)s
2fL]/(( 1-np/npq)
3dp ),
Substituting for nv and nk, gives us our detailed dimensional manifestation of the QFFM:
P = 256( np/npq)
2sL + ( np/npq )s
2fL (6)
3(1-np/npq)
3dp
2
2(1-np/npq)
3dp
Where s = the superficial fluid velocity (fluid flux).
Page 17 of 23
423
Quinn, H. M. (2023). A Fluid Dynamic Development Like None Other. European Journal of Applied Sciences, Vol - 11(2). 407-429.
URL: http://dx.doi.org/10.14738/aivp.112.14344.
We define s = 4q/(D2) and, substituting this into equation (6), gives:
P = 1024(np/npq )
2qL + (np/npq)q2fL (7)
3(1-np/npq)
3dp
2 D2
(1-np/npq)
3dpD4
Re-expressing equation (7) as the pressure gradient, gives:
P = 1024(np/npq)
2q + (np/npq)q2f (8)
L 3(1-np/npq)
3dp
2 D2
(1-np/npq)
3dpD4
Finally, substituting for , and dp gives:
P = 1024(np/npq)
2q + ( np/npq)q2f (9)
L 3(-np/npq)
3(dpmp)
2 D2
(-np/npq)
6(dpmp)D4
We note here that equation (9) contains some remaining dependent variables, i.e., npq, . We
choose not to substitute for these labels because using only independent variables in the
resultant equation would render the formula too cumbersome to write because of the indices
involved in some of the terms, e.g., (1-np/npq)
3. We emphasize here, also, that equation (9) is
based upon measured values to define the particle fraction within the packed conduit and that
the value of np, the number of particles of diameter dp, reconciles all the free space within the
packed conduit.
Note of Clarification: Because a practitioner can only measure the total pressure drop across a
given packed conduit under study, since pressure measuring devices cannot distinguish
between viscous and kinetic contributions to pressure drop, equation (5) has been the topic of
discussion in most all conventional fluid models. Let us therefore restate the viscous term and
compare it to that of conventional wisdom. Substituting for , we can see that 256/3 = 268
approx. represents the viscous constant in the QFFM. This compares to the value of 180 in the
case of the Kozeny/Carman model and 150 in the case of the Ergun model, both of which
underestimate the viscous contributions. This, of course, means that, by default, these models,
simultaneously, overestimate the kinetic contributions since total pressure drop is the sum of
the constituent parts. These are, in part, the specific short comings of these models which we
have referred to above. Conversely, we point out that a value of 270 for this term has been
published by Giddings in his 1991 text book [13], a value very close to that of the QFFM.
Giddings’ development of his own equation containing this value was the subject of a detailed
study previously by this author [13].
An Empty Conduit-A special Case
The QFFM is a universal theory that embraces a closed conduit packed with solid particles, as
the general case, and a conduit packed with particles of free space, as a special case. In scientific
terms, we define solid particles, as particles having a particle porosity, p, of less than unity, i.e.,
0 ≤ p < 1, and particles of free space, as particles having a particle porosity of unity, i.e., p = 1.
In laymen terms, this means the general case, on the one hand, embraces both nonporous
particles, i.e., particles which are solid throughout, and particles which are partially porous, i.e.,
they have a solid skeleton but they possess internal pores, but the special case, on the other
hand, embraces only particles which are totally porous, i.e., they have no solid skeleton.
Page 18 of 23
Services for Science and Education – United Kingdom 424
European Journal of Applied Sciences (EJAS) Vol. 11, Issue 2, April-2023
Let us now adjust our general equation in the QFFM, to accommodate our special case. We
accomplish this by imposing a limiting boundary condition on our general formula, as follows:
Let dp = D, and let np = -npq. Consequently, we can now state the modified dimensional equation
in the QFFM, which applies to empty conduits as:
P = 128q + q2f (10)
L 3D4 248D5
THE QFFM SOLUTION FOR PERMEABILITY
We shall now use the QFFM to solve the Conservation Laws, a.k.a, The Laws of Continuity. To
do this, we will execute the QFFM, in reverse, as it were. By, “in reverse”, we mean that we will
begin with the results of a permeability experiment in a packed conduit, and use the QFFM to
compute (back-calculate) the input variables. This process is shown in Table 1 in steps
numbered 75 to 82.
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 [16]. We can write the Forchheimer equation as
follows:
j = as + bs
2 (11)
Where, a, and b, are the Forchheimer coefficients for the viscous and kinetic contributions,
respectively.
Thus, we can see from equation (11) that hydraulic conductivity is a quadratic function of fluid
flux. It is customary in engineering circles to make a plot of equation (11), a typical example of
which is shown in Figure 8.
Page 19 of 23
425
Quinn, H. M. (2023). A Fluid Dynamic Development Like None Other. European Journal of Applied Sciences, Vol - 11(2). 407-429.
URL: http://dx.doi.org/10.14738/aivp.112.14344.
Figure 8. Hydraulic conductivity as a quadratic function of fluid flux.
As shown in Figure 8, 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. This data is to be found in a paper by Gritti et al. [17].
Since the QFFM 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:
a = 4rh
3 (12)
3fgdc
2
b =
(13)
2gdc
It follows from equation (12) above that we may write:
= 3afg (14)
dc
2 4rh
3
Similarly, it follows from equation (13) above that we may write:
= 2bg (15)
dc
Therefore, in order to solve the pressure/flow equation, we must satisfy both equations (14)
and (15) simultaneously.
From equation (14), let us assume that:
= ()
dc
2
From equation (15), let us assume that:
= ()
dc
Let us further assume that:
x = ()
Similarly, let us assume that:
y = ()
Page 20 of 23
Services for Science and Education – United Kingdom 426
European Journal of Applied Sciences (EJAS) Vol. 11, Issue 2, April-2023
It follows that we may now write the permeability solution for closed conduits as:
dc = 1 (20)
x
(1/6)y
(1/2)
= x
(1/6)
(21)
y
(1/2)
The above simultaneous solution for the values of dc and depends, not only, upon the
independent variables identified above, but also, upon the value of in equation
(13) However in turn, depends upon the value of other variables including dc, a dependent
variable itself and, accordingly, and problematically, this is the conundrum of solving the
Pressure/Flow equation. Furthermore, the independent variable np (number of particles), 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. Additionally, there is no more
sensitive relationship in all of physics between the value of np in the Forchheimer coefficient b, and
the value of the pressure gradient P/L, when the fluid flow profile contains significant kinetic
contributions.
THE FLOW PROFILE-A HARMONIC OSCILLATOR
Steps # 54 through 74 in Table 1, herein, describes the fluid motion within the HQC as that of
damped harmonic motion. This is the only fluid flow model which teaches this type of fluid
motion within a packed conduit. The mathematics underlying the fluid motion is well
documented/understood in the engineering literature and, accordingly, does not require
additional explanation here. The characteristics of the motion are on display in our Fig 1 and
Fig 2 herein.
QFFM VALIDATION
We include as validation for the QFFM several published classical studies representing both
packed and empty conduits, in our Fig 9 and Fig 10 herein. Both plots are presented as equation
(1), i.e., Quinn’s Law.
Page 21 of 23
427
Quinn, H. M. (2023). A Fluid Dynamic Development Like None Other. European Journal of Applied Sciences, Vol - 11(2). 407-429.
URL: http://dx.doi.org/10.14738/aivp.112.14344.
Fig 9
Fig 10
As shown in both Fig 9 and Fig 10, the validation is displayed over 11 orders of magnitude of
the modified Reynolds number which includes all fluid flow regimes of laminar through
transitional and fully turbulent. Note also that the data sets in the legends represented by HMQ- 1, HMQ-2 and HMQ-4 are homegrown experiments carried out by the current author, the details
of which are to be found in reference [1] herein. The data represented by Farkas et al. is to be
found in reference [17] and that of Giddings in reference [18].
CONCLUSIONS
In this paper we have presented the QFFM as a novel development relative to fluid dynamics in
closed conduits. The major conclusions to be drawn from this paper are as follows:
1. This model accommodates both packed and empty conduits, seamlessly.
2. It demonstrates that the term, np, the number of particles present in a packed conduit of
diameter dp, is a necessary independent variable in the pressure flow relationship.
3. It demonstrates that the term, ep, the particle porosity is, also, a necessary independent
variable in the pressure flow relationship.
4. It provides the ability to evaluate the compressibility of particles, using permeability
measurements, by virtue of its ability to identify a changing value for the particle
porosity term ep.
5. It enables the validation of the model over 10 orders of magnitude of the modified
Reynolds number.
6. It demonstrates that the fluid flow profile in packed conduits is best characterized as
damped simple harmonic motion.
7. This model renders the Moody Diagram obsolete since it provides exact computations
over the entire fluid flow regime, including the turbulent regime, thus, eliminating the
subjectivity of interpreting plotted look-up curves.
Page 22 of 23
Services for Science and Education – United Kingdom 428
European Journal of Applied Sciences (EJAS) Vol. 11, Issue 2, April-2023
8. In the case of an empty conduit, there are 7 packed conduit variables, which have a
constant value, which confirms the former as a special case of the latter.
Finally, this new fluid flow model puts an end, once and for all, to the notion that turbulent flow
is driven by chaos. Rather, it confirms that fluid flow in closed conduits is highly structured in
nature and subject to exact scrutiny when a sufficient number of data points are measured to
accommodate the period of motion, for any fluid flow embodiment under study.
Conflict of Interest.
The author has no conflict of interest in this publication either financial or otherwise.
References
[1] P. C. Carman, “Fluid flow through granular beds,” Transactions of the Institution of Chemical Engineers,
vol. 15, pp. 155–166, 1937
[2] S. Ergun, Fluid Flow Through Packed Columns, Chem. Eng. Progr. vol. 48, pp. 89-94, 1952
[3] J.L.M. Poiseuille, Memoires des Savants Etrangers, Vol. IX pp. 435-544, (1846); BRILLOUIN, M. (1930) Jean
Leonard Marie Poiseuille. Journal of Rheology, 1, 345.
[ 4] H. Blasius (1908). "Grenzschichten in Flüssigkeiten mit kleiner Reibung". Z. Angew. Math. Phys. 56: 1–37
[5] H. M. Quinn, Quinn’s Law of Fluid Dynamics; Pressure-driven Fluid Flow through Closed Conduits. Fluid
Mechanics. Vol. 5, No. 2, 2019, pp. 39-71. doi: 10.11648/j.fm.20190502.12
[ 6] R. B. Bird, W.E. Stewart, E. N. Lightfoot, Transport Phenomena, John Wiley & Sons, Inc., p. 190, 2002
[ 7] B.J. Mckeon, C.J. Swanson, M.V. Zagarola, R.J. Donnelly and A. J. Smits. Friction factors for smooth pipe flow;
J.Fluid Mech. (2004), vol. 511, pp.41-44. Cambridge University Press; DO1;10.1017/S0022112004009796.
[8] B.J. Mckeon, M.V. Zagarola, and A. J. Smits. A new friction factor relationship for fully developed pipe flow;
J.Fluid Mech. (2005), vol. 238, pp.429-443. Cambridge University Press;
DO1;10.1017/S0022112005005501
[9] C.J. Swanson, B. Julian, G. G. Ihas, and R. J. Donnelly. Pipe flow measurements over a wide range of Reynolds
numbers using liquid helium and various gases. J. Fluid mech. (2002), vol. 461, pp.51-60. Cambridge
University Press; DO1;10.1017/S0022112002008595
[ 10] J. Nikuradze, NASA TT F-10, 359, Laws of Turbulent Flow in Smooth Pipes. Translated from
“Gesetzmassigkeiten der turbulenten Stromung in glatten Rohren” VDI (Verein Deutsher Ingenieure)-
Forschungsheft 356.
[ 11] J. Nikuradze, NACA TM 1292, Laws of Flow in Rough Pipes, July/August 1933. Translation of
“Stromungsgesetze in rauhen Rohren.” VDI-Forschungsheft 361. Beilage zu “ Forschung auf dem Gebiete
des Ingenieurwesens” Ausgabe B Band 4, July/August 1933.
[ 12] T. Buchwald, G. Schmandra, L. Schützenmeister, T. Fraszczak, T. Mütze, U. Peuker, Gaseous flow through
coarse granular beds: The role of specific surface area. Powder Technology 366 (2020) 821–831
[13] J. C. Giddings, Unified Separation Science, John Wiley & Sons (1991)
[ 14] H. M. Quinn, Reconciliation of Packed Column Permeability Data-Part 1. The Teaching of Giddings
Revisited, Special Topics & Reviews in Porous Media-An International Journal 1 (1), (2010) 79-86
Page 23 of 23
429
Quinn, H. M. (2023). A Fluid Dynamic Development Like None Other. European Journal of Applied Sciences, Vol - 11(2). 407-429.
URL: http://dx.doi.org/10.14738/aivp.112.14344.
[15] P. Forchheimer, Wasserbewegung durch boden. Zeit. Ver. Deutsch. Ing 45, 1781–1788 (1901)
[16] F. Gritti , D. S. Bell , G. Guiochona; Particle size distribution and column efficiency, An ongoing debate
revived with 1.9 _m Titan-C18 particles; Journal of Chromatography A, 1355 (2014) 179–192
[ 17] T. Farkas, G. Zhong, G. Guiochon, Validity of Darcy’s Law at Low Flow Rates in Liquid Chromatography
Journal of Chromatography A, 849, (1999) 35-43
[ 18] J. C. Giddings, Dynamics of Chromatography, Part I: Principles and Theory, Marcel Dekker, New York, NY,
USA, 1965