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European Journal of Applied Sciences – Vol. 11, No. 3

Publication Date: June 25, 2023

DOI:10.14738/aivp.113.14714

Quinn, H. M. (2023). The Fluid Dynamics of Conduit Hydrodynamic Entrance Effects Explained: A Rebuttal Paper. European Journal

of Applied Sciences, Vol - 11(3). 649-660.

Services for Science and Education – United Kingdom

The Fluid Dynamics of Conduit Hydrodynamic Entrance Effects

Explained: A Rebuttal Paper

Hubert M Quinn

Department of Research and Development,

The Wrangler Group LLC, 40 Nottinghill Road, Brighton, Ma.02135, USA

ABSTRACT

Fluid flow in closed conduits is a field of study that has been characterized in the

published literature, up until recently, by contradictions, false assertions and

sometimes even more egregious misrepresentations of what the Laws of Nature

dictate. The issue of conduit hydrodynamic entrance effects, in particular, has been

studied and published on, extensively, over the last 100 years but this endeavor has

yielded more heat than light. A recent publication (2023) appeared in the

prestigious journal of Fluid Mechanics entitled, “Hydrodynamics of finite-length

pipes at intermediate Reynolds numbers”, in which the entire paper was dedicated

to this oftentimes misunderstood phenomenon. We challenge the author’s

conclusions, herein, and in so doing, we demonstrate that their measured data is

fully consistent with the recently published “Quinn’s Law of Fluid Dynamics in

Closed Conduits” and, consequently, we provide an alternative explanation which

establishes a comprehensive understanding of the physics underlying the so-called

conduit hydrodynamic end-effects, a necessary consideration in this study, but one

which the authors surprisingly chose to ignore. Furthermore, we validate our

analysis in the context of the two classical published studies of (a), Nikuradze, for

his sand-roughened pipes, and (b), the Princeton Super Pipe, for smooth walled

pipes, both of which are considered contemporary gold standards in the field of

fluid dynamics in closed conduits.

Keywords: pipe flow boundary layer, transition to turbulence, pipe flow, Wall Effects,

Pipe Entrance Effects.

INTRODUCTION

This paper is a rebuttal to the publication appearing in the Journal of Fluid Mechanics

(Pomerenk et al 2023). Since this rebuttal is based upon an analysis which, in turn, is based

upon the recently published “Quinn’s Law of Fluid Dynamics” (Quinn 2019,2019,

2020,2020,2020,2023), we will briefly explain why the Quinn Fluid Flow Model (QFFM) is

superior to any current extant fluid flow model when considering conduit hydrodynamic

entrance effects. This is because the QFFM contains an expression for the Pressure/Flow

relationship like none other, to the extent that it is universally valid over the entire fluid flow

regime. In addition, it is equally valid for conduits packed with particles and, accordingly,

contains terms for all elements of hydrodynamic wall-effects which include, the viscous

boundary layer, wall roughness and entrance-effects. Additionally, it conforms to the teachings

of the Forchheimer fluid model (Forchheimer 1901), regarding the relationship between fluid

velocity and pressure drop outside of the laminar flow regime. We recommend that the reader

look to the original publications of the QFFM contained in the references herein for the details

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European Journal of Applied Sciences (EJAS) Vol. 11, Issue 3, June-2023

of all the relevant governing equations. Thus, we can write the teaching of the QFFM in its

dimensionless form as;

PQ = k1 + k2CQ (1)

Where, PQ represents dimensionless pressure drop, k1 and k2 are universal constants and CQ

represents the dimensionless fluid flow term.

In order to make it more convenient for the reader, we include herein our Table 1, which

contains the nomenclature and governing equations of the QFFM.

Table 1 Nomenclature and Governing equations

THE WALL EFFECT TERM l

We define the term CQ = lQN where both l and QN are dependent variables. It will be appreciated

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

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651

Quinn, H. M. (2023). The Fluid Dynamics of Conduit Hydrodynamic Entrance Effects Explained: A Rebuttal Paper. European Journal of Applied

Sciences, Vol - 11(3). 649-660.

URL: http://dx.doi.org/10.14738/aivp.113.14714

terml deals with the wall effect. We can re-express equation (1) to identify the role of l as

follows:

PQ = k1 + k2lQN (2)

We should point out, at this point, that conduit hydrodynamic entrance effects are a subset of

the overall wall-effect phenomenon, but this term is typically omitted from most general

equations because, in practice, a practitioner will always choose a sufficiently long conduit to

ensure that entrance effects are eliminated. This, in turn, makes sense since entrance effects

typically detract from the overall conduit performance and add nothing of value for the

practitioner. In the case of chromatographic columns, for instance, end-effects totally destroy

the ability of the packed conduit to do meaningful separations. This means that conduit end- effects are more of an academic interest than anything a typical practitioner is interested in.

Nonetheless, the QFFM being a rigorous theoretical development based upon the fundamentals

of the physics underlying fluid flow in closed conduits, easily accommodates this element of

wall effect and, in so doing, teaches that there are just five elements to the wall effect:

1. primary wall effect (W1);

2. secondary wall effects ( W0 and W2);

3. residual secondary wall effects (W0R and W2R);

4. net wall effect (WN) and finally,

5. wall normalization coefficient (l).

The net wall effect is the sum of the primary and residual secondary wall effects, WN = (W1 +

W0R +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 at

very low flow rates (laminar flow).

The wall normalization coefficient l = (1 +WN), and enters the pressure flow relationship

through the kinetic term. Accordingly, the l 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 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 b0 = (k1/k1 +k2QN), where

k1 = (64p/3) and k2 = (1/8p), and are the universal constants displayed in equation (1) above.

QN = dRem is the Quinn number. The symbol Rem stands for the well-known modified Reynolds

number and d = 1/(1-np/npq)3.

Secondly, the formula for the primary wall effect is W1 = (b0(1/3)/ t). It is, therefore, also a

derivative of the tortuosity factor, t = dg where g = (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.