Establishing The Boundaries of Our Ignorance: An Exposition into Single-Phase Closed Conduit Flow

A mixture of two or more flowing phases in a stationary bounding medium occurs ubiquitously in various industries and is of considerable interest and importance. Closed conduit (or pipe) flow is a branch of fluid mechanics that represents one kind of multiphase flow in which one phase (the bounding medium at rest) completely encloses the adjoining phase or phases flowing through it. Bounded flows in which the adjoining flowing phases are not completely enclosed by a closed conduit represent another kind of multiphase flow (e.g. flow past a finite body, action of wind on ocean waves, etc.). Examples of closed conduits include pipes, channels, ducts, enclosed passages and tubes. The adjoining phases, flowing simultaneously in a multiphase mixture, can be any combination of vapor-like, liquid-like and solid-like substances.

In the general sense, pipe flow will include a set of scenarios in which one or more of the phases present do not flow. The phases that do not flow are the main source of the velocity gradients (and thus momentum transport, kinetic energy dissipation and flow regimes) in the adjoining phases that do flow. This is because the transport of quantities like momentum and energy occur in the direction from points in the flow field where velocity is large to those where it is small. We call this a flow relative to no-flow scenario. Indeed, a single-phase pipe flow is one special case of the general multiphase pipe flow problem – the fluid being a moving phase continually exchanging momentum and energy with a stationary phase, the pipe. In Bird et al., 2002 (or BSL), the multiphase language used to describe this scenario is 'interphase transport', meaning the macroscopic transport of mass, momentum, energy and entropy between a flowing phase within a system and a bounding surface of that system (usually a stationary phase). Some investigators share this generalist view that, without further specification, multiphase closed conduit flow encompasses nearly all of fluid mechanics (e.g., Prosperetti and Trygvasson, 2007). In contrast, other investigators view multiphase closed conduit flow as a field that is distinct from applied fluid mechanics (e.g., Theophanous and Dinh, 2003).

From the preceding discussion, specification is clearly necessary in multiphase closed conduit flow. In the sense in which this field of study is practically encountered at the industrial level in terms of averaged descriptions, it is defined by a macroscopic system comprising two or more flowing phases which are completely enclosed by a stationary solid-phase (the boundary that is the pipe). In this article, we explore the fundamental derivations underlying macroscopic single-phase pipe flow fluid mechanics concepts up until the limits of our boundaries of ignorance.

1. The Genesis: Phenomenological Transfer Laws

In the balance equation for the rate of change of axial-momentum in single-phase pipe flow, there is one unmeasured system quantity – the wall friction (or viscous) loss factor. Understanding this loss factor is, therefore, where we must start. We start here since it demarcates the ignorance-boundaries of the single-phase pipe flow problem. Indeed, if a multiphase pipe flow problem is well understood, then the best indicator to infer this will be that the accuracy of its solutions will hopefully approach the corresponding single-phase flow accuracy. In a general sense, however, the accuracy of any multiphase pipe flow scenario will always be lower-bounded by the maximum of the corresponding single-phase flow accuracy and multiphase flow instrumentation error.

Now, the general form of the phase-to-boundary phenomenological transfer laws for any conservable or non-conservable quantity, Y, is expressed as shown in Eqn. 1 below. For example, a familiar example of Eqn. 1 is the transfer law for thermal energy (or heat), as shown in Eqn. 2 below.

In estimating viscous loss, the wall friction loss factor is defined as the proportionality constant relating the rate of irreversible kinetic energy decrease due to the presence of the stationary pipe wall and the relative rate of change of kinetic energy, which, in the form of Eqn. 1 above , can be stated as shown in Eqn. 3 below.

2. The Closure: Relation to Friction Factor and Momentum Flux Transfer Area

As seen in Eqn. 3 above, the wall friction loss factor is the irreversible kinetic energy transfer coefficient between the flowing phase and its stationary boundary. Note that the notation, terms and conventions of BSL are used for simplicity. For a straight conduit with cross-sectional area open to flow, A, and length, ΔL, the wall friction loss factor is given in BSL in terms of a mean hydraulic diameter (D_subscript_H), defined as:

A deeper understanding of mean hydraulic diameter as revealed in my most recent #ALRDC presentation (download here) reveals that the reason WHY the wall friction loss factor is defined in this way - that is - this fundamental, analytical definition allows us to re-write Eqn. 3 in the general form of the phase-to-boundary phenomenological transfer law (Eqn. 1), as shown in Eqn. 5 below.

We now see that the mean hydraulic diameter-based definition of wall friction loss factor in Eqn. 4a above brings in the dependence of the phase-to-boundary mechanical (kinetic) energy flux on the laminar, transitional and turbulent flow regimes of single phase flow. This is an important step. In other balance equations, such as the continuity or momentum equations, it is of no consequence whether the flow regime is laminar or transitional or turbulent, but for the mechanical and thermal energy equations, these regimes are important. As Bobok (1993) points out, different formulations of the mechanical energy balance equation can be written for turbulent flow, based either on time-averaged velocities or on the actual velocity fluctuations. Also, note the rate at which the phase mechanical energy decreases (or viscously dissipates) due to the presence of the stationary pipe wall is related through Eqn. 5 to quantities that are subject to direct experimental observation.

Apart from bringing in the dependence on laminar and turbulent flow regimes, the mean hydraulic diameter-based definition of wall friction loss factor in Eqn. 4a additionally defines the surface area over which the flux of mass, momentum, energy or entropy is transferred, i.e. ZΔL. This has an importance all on its own aside from the wall friction factor. The key insight is that, at any given axial location in a pipe, the interface between the two phases (the flowing fluid phase and the stationary pipe phase) is localized with regard to cross-sectional position – it is always at the boundary of the flow field for closed-conduit flows. This interface defines the averaged momentum flux transfer surface between the flowing phase and its stationary boundary (the pipe wall). Therefore, the Fanning friction factor can itself be interpreted as the quantity that directly relates the viscous loss factor with the ratio of the two most important surfaces bounding the flowing phase, ZΔL, and, A.

Next, it is vital to note that at any given axial location in a pipe, the area open to flow, A, is the area that cancels the area defining the mass flow rate of the phase in Eqn. 3. The result of this cancellation is that the right side of Eqn. 5 does not contain the term, A. In general, the area open to flow, A, is unrelated to the actual area occupied by the phase. The key insight is that in single-phase closed conduit flow, it just so happens that these areas coincide, but in the multiphase flow scenario, they do not. In either scenario, the area open to flow, A, is the area that defines the mass flow rate of the flowing phase and possesses no other meaning.

3. The Generalization: Boundary Force

In terms of the actual, measurable boundary force (i.e. the force the flowing phase exerts on the conduit wall), this force can be found by equating the axial force and mechanical energy gradient balance equations, to yield Eqn. 6 below.

Then, combining Eqn. 6 and 5, we can re-state the phenomenological transfer law for the phase-to-boundary mechanical energy flux in Eqn. 5 in terms of the phase-to-boundary momentum flux, as shown in Eqn. 7 below.

As seen above, Eqn. 7 is in the classic form of a phenomenological transfer law with flux transfer coefficients in front of relative quantity-concentrations. This is the form that makes it possible to see the generalization of transport phenomena as exemplified in BSL. Nonetheless, it is instructive from a pedagogical view to show this same equation in a form that may be more familiar to the subset of investigators who prefer to interpret friction factor as a mechanical (kinetic) energy dissipation coefficient, as shown in Eqn. 8 below.

4. The Scaling: Microscopic Equations of Change and Macroscopic Balance Equations

Now that a mathematically self-consistent expression is in place for the boundary force (Eqn. 7), the next basic question concerning the hydrodynamics of the flow, is to determine how this quantity contributes to the axial-momentum balance (and therefore, pressure gradient) in pipe flow? The answer to this question is not trivial. In particular, two very important and related sub-questions arise:

(1) How is the boundary force, a macroscopic (global) quantity, related to the microscopic (local) equation of change for axial-momentum?

(2) Is the equation of change for axial-momentum the only local conservation equation that should be used for defining this relationship?

As is evident from the averaged flow literature, there are several alternative approaches and answers to these very basic questions. In some cases, macroscopic and microscopic quantities are often mixed, in which case partial differential equations no longer describe microscopic (local) flow quantities, but rather, their time- and space-averaged descriptions. In contrast, this side-by-side mixing of microscopic and macroscopic quantities are absent in classical analyses of fluid mechanics (e.g. Whitaker, 1982, Landau and Lifshitz, 1987). In BSL, there is a clear separation of what constitutes microscopic equations of change within the flow system and what constitutes macroscopic balance equations when these equations of change are integrated over the entire volume of the flow system (Bird, 1957; Slattery and Gaggioli, 1962), as represented in Eqn. 9 below.

Now, although this separation of microscopic and macroscopic quantities may seem to some pipe flow investigators a matter of semantics, the separation of these quantities is very important, as shown in BSL. While the pipe flow investigators who do not separate these quantities have clear reasons for doing so (e.g., they may want to enforce property spatial variations to be allowable only in the axial direction in their partial differential equations), the key realization arising from this separation is that some terms will show up only at the macroscopic (or integral) level. This means that there will be some terms in the balance equations that are different from the terms in the equations of change. This is because the phase-to-wall (or interphase) flux terms in the balance equations describe the transfer processes at the boundary, whereas the flux terms in the equations of change describe the transport processes within the main stream (e.g. conduction, convection). This, of course, has dramatic mathematical modeling implications in either single-phase or multiphase pipe flow, namely, that the averaging method employed for the net accumulation, transport and source terms of the local equations of change does not apply to the separate (and usually empirical) net transfer terms at the boundary. Specifically in multiphase pipe flow, this means that: (a) before averaging is performed, the phase function, as explained in Drew (1983), is multiplied by only the terms in the local equations of change, and (b) after averaging the local equations of change, the multiphase wall flux transfer terms that appear in the macroscopic balance equations, have no fundamental basis for being phase function-weighted terms.

Additionally, we note that the general phenomenological forms of the net transport terms in the local equations of change are different from the general form of the net transfer terms in the integral balance equations, shown in Eqn. 1. For example, in most scenarios (though not all), conductive main stream fluxes can be generally expressed Eqn. 10 below.

Fourier’s law of heat conduction in a medium is one example of a phenomenological transport law in the form of Eqn. 10 where Y = phase thermal energy (or heat). As another example, convective main stream fluxes are generally expressed as Eqn. 11 below.

In light of the foregoing discussion, we are now in a solid position to relate the boundary force to the local equation of change for axial-momentum. We start from the equation of motion. As shown in BSL, it is the equation from which the Navier-Stokes equation can be derived. If we assume that the momentum transfer rates associated with the viscous stress are small compared to the fluid pressure forces across the entrance and exit planes of the considered system, we can arrive at a macroscopic balance equation for axial momentum in an arbitrary x-direction, as:

Eqn. 12 displays a deceivingly simple character. It shows the link between the macroscopic terms that are derived from basic principles and the one term that does not have a fundamental basis, the net wall flux transfer term (the last term in Eqn. 12). Analogously, the situation is not different for multiphase pipe flow - there will be net accumulation, transport and source terms derived from basic principles and there will be net interfacial and wall flux transfer terms that are not. So, in general, it is not true to say that the single-phase or multiphase pipe flow balance equations lack a fundamental theoretical basis - this is only true for some of the terms.

Now, if instead of using only the equation of motion to arrive at the formulation above in Eqn. 12, we instead combine the equation of motion with the viscous stress neglected (as before) and the equation of continuity (mass conservation) we will arrive at the familiar Euler’s equation (Landau and Lifshitz, 1987), as seen in Eqn. 13 below.

Note that the substantial time (or Lagrangian) derivative evaluated with the phase velocity in Eqn. 13 above denotes the rate of change of the velocity of a given fluid particle as it moves in space (i.e., its transportive velocity), whereas the partial time derivative denotes the rate of change of the fluid velocity at a fixed point in space, which is the local acceleration/deceleration. Thus, instead of Eqn. 12 above, a new combined-equation-of-motion-and-continuity macroscopic balance equation for axial-momentum in an arbitrary x-direction can be written from Eqn. 13, as:

When BSL notation is used for Eqn. 14, it can be written in terms of Eqn 15 below.

For a straight, constant cross-section pipe, with steady-state flow in the arbitrary x-direction at an angle of θ degrees to vertical, Eqn. 15 becomes (using mass flux, G, in place of velocity) Eqn. 16 below.

Finally, combining Eqn. 16 and Eqn. 7, we arrive at Eqn. 17 below.

Eqn. 17 is the steady-state, single-phase balance equation for the rate of change of axial momentum in pipe flow re-expressed as an axial pressure gradient balance equation. For a very wide class of single-phase pipe flow problems in fluid dynamics, this equation has proven to work quite well. In the field of multiphase pipe flow, Eqn. 17 works so well that it is used by experimenters to test for proper operation and to calibrate measurements in experimental flow loops. Other than experimenters, many multiphase pipe flow modelers consider the single-phase rate of change of axial-momentum balance equation to be “well established” (Heywood and Cheng, 1984) or “firmly established” (Ishii and Hibiki, 2006).

Therefore, the questions that abound concerning Eqn. 17 for any particular single-phase flow scenario, by necessity, must also abound for the corresponding multiphase flow scenario. This is the reason that Eqn. 17 is the most logical place to start an analysis of multiphase pipe flow - it defines our ignorance boundary. As Geoffrey Hewitt and Joseph Kestin point out, “Even single-phase turbulent flows still defy prediction from a fundamental point of view” (Hewitt, 1983), and, “Even one-phase flows are not totally on a sound footing because turbulence is still untractable” (Kestin, quoted in DiPippo, 1980). Tom Mullin reminds us that even as basic a question as transition to turbulence in single-phase pipe flow remains unsolved: “It is an enigma as all theoretical and numerical evidence suggests that the base state of fully developed flow, Hagen-Poiseuille flow, is linearly stable. The transition to turbulence is abrupt, mysterious, and largely dependent on the quality of the facility used in any experimental investigation. It is therefore not an example of transition via a sequence of instabilities or bifurcations” (Mullin, 2011).

5. The Caveats: Different Formulations and Interpretations

Convective Acceleration or Deceleration

A caveat regarding Eqn. 17 is that in the majority of flow loop scenarios (e.g. unobstructed, constant cross-sectional area, moderate flow rates or steady-state flow), the convective acceleration/deceleration term has not been tested in any serious way because its contribution was almost always, by design, minimal. In some cases, pipe flow investigators even acknowledge this inherent design in their descriptions of reported experiments. Of course, the convective acceleration/deceleration term becomes quite important in high rate flows at relatively low pressures (e.g. flows subject to near atmospheric pressures). Other cases where the convective acceleration/deceleration term can become crucial are obstructed, varying cross-sectional area, high flow rates or transient flows involving rapid rate changes. As noted in prior investigations (e.g., Majiros and Dukler, 1961, Silvestri, 1964), in both single-phase and multiphase flow scenarios, the contribution of this acceleration/deceleration term can be a substantial fraction of the total pressure gradient, especially in the case where vapor densities will vary considerably along the flow direction because of the relatively high variation of the line pressure (e.g., as in lower pressure systems). Nevertheless, it is important to recognize that this term is erroneously considered unimportant by some investigators and is crudely modeled (if at all) in both single-phase and multiphase flow scenarios as a result of this preconceived belief.

Wall Friction Factor

The most important thing to note about Eqn. 17 is that the dependent variables are related to each other as well as to known system parameters in this one equation through one unknown - the wall friction factor - which is the only quantity that cannot be directly measured. Also, note that the wall friction factor is not the only thing that is unknown because there was a series of assumptions invoked in the going from the local to the integral level. However, unlike the wall friction factor, all of the other quantities in Eqn. 17 can be carefully (though not easily) measured and therefore it is possible to falsify or validate any of the individual assumptions or hypotheses that led to it for a given friction factor. This is why the better pipe flow experiments (and models) enforce that the friction factor not be changed from its best tractable value for the scenario under consideration. For example, in flow loops, hydraulically smooth pipes are often used to force the behavior of friction factor to its best predictable state unless, of course, the departure from hydraulic smoothness is itself under investigation, e.g., the study by Chisolm and Laird (1958) for multiphase flow or the Superpipe experiments for single phase flow.

Existence of Different Formulations

It is instructive (and insightful) to carefully observe the familiar path taken to arrive at Eqn. 17 - that is - starting from the local (microscopic) conservation equations to eventually arrive at a global (macroscopic) momentum balance equation. Classical fluid dynamics treatments like Landau and Lifshitz (1987) and BSL clearly informs us of how other balance equations (e.g., the mechanical energy equation and its approximated form, the engineering Bernoulli equation) can be obtained from the equation of motion and what specific sets of assumptions are needed to arrive at them.

Nevertheless, there are some pipe flow investigators that do not follow this classical (correct) path and, instead, create their own interpretation of the mechanical energy balance equation as though it is derived from “extending” or “converting” the total energy balance equation. In this latter interpretation, closed-system, homogeneous (pure component), equilibrium thermodynamics relations are combined in an ad hoc way with the flux relations across the boundaries of a control volume described by an open system total energy balance equation. In rare cases, even multiphase pipe flow investigators will base their open-system, multiphase, multi-component analyses on this interpretation. In these exceptional cases, investigators will claim a 'general mechanical energy balance' for their multiphase flow by introducing various artificial definitions and mixing rules for different variables in their derived mechanical energy balance equation (e.g., mixture viscosity, mixture friction factor, and so on). One may even find among these exceptional cases 'multiplier factors' that appear with their kinetic energy terms that take on different adjustable values for different scenarios, both in the single-phase and multiphase mechanical energy balance equations (sometimes referred to as a 'velocity profile correction term').

It must be recognized that there is nothing general or fundamental about introducing artificial variables, i.e., variables that can only be inferred and not measured. These rather peculiar developments are in fact different total pressure gradient correlations depending on the choice of mixing rules and definitions for the different variables. I emphasize here the following BSL quote (pg. 458) as being crucial to the correct understanding of the basic equations in single-phase flow:

“The mechanical energy balance is not ‘an alternative form’ of the energy balance”

Existence of Different Interpretations

Other than the core equations derived above, a lack of fundamental understanding of the core concepts in single-phase pipe flow sometimes gets extended to multiphase pipe flow. A concrete (and historically traceable) example that depicts this unfortunate fact is how some investigators misconstrue what irreversible energy conversion means in multiphase pipe flow in presence of upward and downward pipeline elevation changes.

6. Summary

It is put forward in this article that we cannot approach the modeling the hydrodynamic multiphase pipe flow problem without understanding and carefully demarcating the ignorance boundaries of the corresponding single-phase pipe flow problem. I've shared some deeper insights into some important terms and derivations of the single-phase closed conduit macroscopic balance equations above. The next fundamental question is: What changes when other flowing phases are introduced into the flow field?

Now that's something to ponder about!

Anand

#deepdivechallenge

References

Bird, R. B., Stewart, W. E., Lightfoot, E. N.: Transport phenomena, 2nd ed., John Wiley and Sons (2002)

Prosperetti, A., Trygvasson, G.: Computational methods for multiphase flow, Cambridge University Press, Cambridge (2007)

Theofanous, T. G., Dinh, T. N.: On the prediction of flow patterns as a principal scientific issue in multi-fluid flow, Multiphase Sci. and Tech., v. 15, no. 1-4, pp. 57-64 (2003)

Bobok. E.: Fluid mechanics for petroleum engineers, Dev. in Petroleum Science, Ser. 32, Elsevier (1993)

Whitaker, S.: Laws of continuum physics for single-phase, single-component systems, Handbook of Multiphase Systems, pp. 1-5 to 1-35 (1982)

Landau, L. D., Lifshitz, E. M.: Fluid mechanics, 2nd Ed., Course of Theoretical Physics, v. 6, Elsevier (1987)

Bird, R. B.: Chem. Eng. Sci., v. 6, pp. 123-131 (1957)

Slattey, J. C., Gagglioli, R. A.: Chem. Eng. Sci., v. 17, pp. 893-895 (1962)

Drew, D. A.: Mathematical modeling of two-phase flow, Ann. Rev. Fluid Mech., v.15, pp. 261-291 (1983)

Heywood, N. I., Cheng, D. C. H.: Flow in pipes, part 2 – multiphase flows, J. Phys. Technol., v. 15, pp. 291 (1984)

Ishii, M., Hibiki, T.: Thermo-fluid dynamics of two-phase flow, Springer (2006)

Hewitt, G. F.: Two-phase flow and its applications – past, present, and future, J. Heat Transfer Engineering, v. 4, no. 1, pp. 67-79 (1983)

DiPippo, R.: The next steps in two-phase flow – executive summary, U.S. Dept. of Energy workshop/symposium, Brown University, Report No. GEOFLO/8 (1980)

Mullin, T.: Experimental studies of transition to turbulence in a pipe, Ann. Rev. Fluid Mech., v. 43, pp. 1-24 (2011)

Majiros, P. G., Dukler, A. E.: Develop. Mechanics, v. 1 (1961)

Silvestri, M.: Fluid mechanics and heat transfer of two-phase annular-dispersed flow, Adv. in Heat Transfer, v. 1, pp. 355-446 (1964)

Chisolm, D., Laird, A. D. K.: Trans. AIME, v. 80, pp. 276 (1958)

#singlephaseflow #multiphaseflow #pipeflow #equationsofchange #balanceequations #BSL #boundariesofignorance