Constain JWR

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Research article/Journal of Water Research.

A new mean velocity equation: a missing l ink betweenmass transport and hydraulics in water quality studies.

Constain A.J.

Researcher, Hydrocloro Tech, Bogot, Colombia [email protected]

ABSTRACT: Following the basic structure of Chezys equation for uniform flow, it is possible todefine a new mean velocity equation involving mass transport parameters instead geomorphologicalones. This new equation allows finding hidden relationships between these two main fields of water sciences, as will be examined in this paper.Double treatment on natural subjects is a common issue in history of science. This is the case of dualitywave-particle in quantum mechanics as was explained by De Broglie, Heisenberg and Dirac, also theduality of mechanics and optics as was developed by Hamilton [1]. In these examples is apparent thatcertain general facts may be seen equally from different sometimes opposite- points of view. Thisfertile idea has led to an enhanced understanding of those subjects that share properties of two differentworlds without contradiction, which appear as a circumstantial not essential- drawback. Then, it may

be interesting to apply the duality concept to some modern problems, not viewed in this way until now,as for example the dynamic of a conservative solute in turbulent, shear flows. To get this aim it isnecessary to explore the relationship between causes and effects in this kind of process: distribution of velocities in first place and dispersion velocity in second place. In this paper it will be examined somethermodynamic aspects of this model to solve it in a proper way, leading to a new equation for meanvelocity of flow as main result. Finally it will be presented an experimental case to verify the accuracyof new model.

Keywords: Hydraulics, mass transport, turbulence.

1. INTRODUCTION

Modern Quality of Water models are crucial tools to handle and planning efficiently several critical processes related with contaminations in channels, streams and other water bodies[2][3].These tools are mainly software packages intended to predict in a scientific basis the different scenariosfor several parameters related with physical, chemical and biological fields of environmental, civil or sanitary concerns, especially on surface flows. It is commonly accepted that a first issue to be analyzedin these topics is the question where the water goes and how water movements affects theconcentration of several entities in flow. This definition is called as the flow and transport issue inwater quality modeling problem. The understanding of the complex relationships between sources of impurities and effects on water quality in moving water bodies is a main task to do these efforts aremainly directed to reveal significant transport and transformation mechanisms for substances of
mailto:[email protected]:[email protected]:[email protected]:[email protected]


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interest, and then to predict the fate of these substances in natural environments. In this way, thesecomplex systems may turn to be a scientific, abstract concept to a down- to- earth engineering aid,solving practical problems.Historically software models as practical tools were possible only until mature development of computer sciences about decades of 1970 and 1980, years in which was possible to put in code complex

numerical methods about conservation laws. In those models the conservation principle may be appliedto all transformation of mass (transport), momentum (flow) and energy (temperature).Usually eachspecific principle has its own set of equations without connections with others, avoiding in some wayan integral, congruent calibration. This drawback arises from the fact that in each system equivalent

parameters fitting equations optimally may have different values. The development of an equation that built a bridge between transport and flow terms will be useful to save time, money and effort usingthese models.

2. RELATIONSHIP BETWEEN MEAN VELOCITY AND DISPERSION VELOCITY

As is well known, in a turbulent flow moving in longitudinal direction there is a strong mix of fluidlayers due to the presence of transverse (and rotational) motions of particles. This possibility (that is not

present in solids) is a consequence of the weakness of interactions in liquid particles that comprisesseveral kinds of motions in small volumes. The final effect of this fact is that advection of different

parcels of liquid has a more uniform distribution of velocity along flow geometry (compared with the parabolic distribution of laminar flow) but with more rapidly changing in time and space point vector velocities. This means that turbulent flows have a higher mixing capability that laminar ones. [4] Figure1.

Figure 1. Mixing capability of laminar and turbulent flow

This very fast changing velocity field in a flow means a more powerful mixing mechanism that inlaminar case. Also means that the heterogeneous nature of molecular motion leads to a greater energylosses. This is because the interaction forces of liquid molecules in such flow have opportunity to berelocated several times ejecting irreversible heat in each cycle, as a macroscopic effect. [5] [6]


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So, it is important to relate mixing effect with the losses in a turbulent flow because they are reflects of the same thing. This means also that there is a thermodynamic explanation of dispersion of liquid (andsolute) particlesin turbulent flows as is shown in Figure 2.

Figure 2 Mixing capability of laminar and turbulent flow

This process is known as shear effect of natural turbulent flows in which each pair of particles in flow isseparated an statistical distance producing a spreading of tracer named dispersion . In this picture,clearly heat Qib is greater than heat Qia , as was explained.Exploring these concepts, one may now to state the ejected irreversible heat as the change of Freeenergy (Helmholtz potential) in the process. This energy was obtained initially from the formationenergy of solute compound.[7]The advantage of this approach is that this is a thermodynamic potentialthat may be write as the work of somedefine driving agent, as for example the osmotic pressure used

by Einstein to write his well-known paper of 1905. [8]. In this paper Einstein showed that the osmotic pressure have not only an associated force exerted in a semipermeable membrane according withclassical theory of ideal gases (vant Hoffs model) but also a driving force that guide the diffusion

process itself. In this picture a solute diffuses in a flow evolving as an ideal gas that spreads spendingenergy (irreversible process). So the free energy supplied by the osmotic force in this process is wastedcompletely as heat ejected to environment. Graphically this concept may be drawing as two oppositeagents that equal each other: the diffusion as a predictable process and the friction that destroy it (bluearrows osmotic effect and red arrows friction action). Figure 2

Then this energetic balance may be stated as follows, for an isothermal system:

S T U S T U F 0 (1)

The U component corresponds to formation energy and then associated with osmotic pressure asdriving force. The TS component corresponds to irreversible heat ejected to environment by internalfriction of flow mechanism, depending more on impulse transfer in molecular mixing of turbulence, asis well known. F has to be zero, according with the totally irreversibility condition of phenomenon. [9]

QibQia


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Figure 2 Two opposite effects: Osmotic pressure force and heat ejection.

Now one may define a set of corresponding works as follows:

f AU d AS T (2)

Considering that these works are developed in identical displacements it is possible to write therequired balance of forces as follows:

d f f f 0(3)

To characterizing ff force in a macroscopic way is necessary to relate it with Ficks diffusion,clearing the mass of liquid taken in account:

t S X c

E M (4)

Here M is the tracer mass, c the tracer concentration S= Z* Y the area element through it passes themass. X is the displacement and V the volume defined as the product S * X . E is the longitudinaltransport coefficient. Then using Newtons definition of force we have the absolute value for osmotic force (using Lagevins concept of virtual force in molecular domains):

t

V X c

E

t X

M f f 2 (5)

By the other hand the friction force may be written regarding that energy losses are proportional to velocity of parcel squared.

Diffusion= Osmotic force

Irreversible heat ejection


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2uk f d (6)

The, integrating in the volume of interest:

t

V X

C E

U k 2 (7)

Here U and C are spatial averages, calculated on cross section of flow. Following the involvedalgebra and clearing the squared mean velocity:

t E

V X C

k U

12 (8)

Taking squared root:

t E V

X C

k U 1 (9)

Now, it is necessary to state that the temporal frame, in which diffusion (osmotic effect) develops, ,is different than the general time frame, t as independent variable. This difference is becausediffusion characteristic time is linked rather with a counting mechanism done by an observer located downstream of solute pouring location. Using a Poissons statistical distribution for thiscounting mechanism, it may be demonstrated that:

t (10)

Here 0.215 if it is used a mean value for distribution as a1.54 as was discovered by TheSvedberg at earlier years of past century [10]

Then, we can rewrite Equ (13) as follows:

E

V X C

k U

22

(11)

Calling 1/ to the first factor it holds:

E

U

21

(12)

This is a new definition for the mean velocity of flow, very similar to Chezys classical equationexcept that instead geomorphologic parameters in this one there is transport parameters.

To search the nature of function, it may be stated as a velocity ratio in the following way:


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UxVdif

(13)

Here the random spreading velocity of solute has a direct definition as the ratio between thecharacteristic displacement, , and the characteristic time, .

dif V

(14)

Accepting the Brownian nature for this spreading effect the above definition may be writing as:

E E

V dif 22

(15)

Then it holds, as we may expect:

E

U 21

(16)

In this way we have now a definition of mean velocity of flow but put in terms of mass transport parameters and not as function of geomorphologic variables, as Chezys equation (17). However, itis interesting to see that both equations have the same quadratic mathematical structure.

RS C U (17)

Then eq. (16) is a dual equivalent of eq . (17) in the same sense that Schrodingers equation is the

dual equivalent of wave equation of mechanics or electromagnetism (Maxwells equations). Eachequation is written in terms of its own set of parameters. Despite the evident differences of theseequations, it is interesting remark that the concept of losses is present in both of them. Chezysequation results as a balance between gravitational and friction agents, meanwhile its counterpartresults as a balance between osmotic pressure and the transport of irreversible heat to environments.

3. SIMILITUDES AND DIFFERENCES BETWEEN TWO MEAN VELOCITY EQUATIONS

It easy to see that the inverse of function is equivalent to Chezys factor, C , that transport drivingagent E is equivalent to Slope (gravitational driving agent), S , and that characteristic limiting factor

, , is equivalent to hydraulic Radius, R. In this sense, eq.(16) is properly time equivalent of Chezysequation put in terms of distance.

But it is possible to go beyond this similitude regarding the expansion of Chezys resistance factor by means of Darcy-Weissbach formula for C:


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f g C

8(18)

In this case f is the friction factor that depends on roughness of surface and on Reynolds number.

Also k is a proportionality losses factor.

V X C

k 21

(19)

As may be noted, f and k play the same thermodynamic role, however first is not dimensional butsecond it is.

It is important to remark that while Chezys equation is defined for uniform flow the Eq . (16) it isnot. This because the shear effect that guide the dispersive transport it is not solely a property of uniform flow but also of a flow in which each elementary velocity in cross section may vary from

point to point. Only it is a requirement that exist a mean value as defined in Eq. (20).

y y

dA z yu A

U ),(1

(20)

The uniformity condition for a flow is a very restrictive one because, strictly speaking it requiresthat every elementary velocity (in each point in plane y*z ) should be equal. This clearly is acondition seldom accomplished by flows, and then the use of Eq. (16) instead Chezys give a higher level of reality to theory. So, accurate application of Eq. (16) is always possible.

4. A HARDWARE-SOFTWARE TOOL TO MAKE MEASUREMENTS WITH NEWTHEORY: RESULTS OF A STUDY CASE.

Authors scientific team in Colombia has developed a special hardware-software tool which mayapply the new formulas explained here. This equipment may operate in in site fashion giving ahuge information of the flow at the same time that the tracer plume is passing by the downstreammeasurement point. The operational routines calculate the special parameters of method as for example: , , etc. With this information it may display a theoretical model superimposed onexperimental curve, and give a table with transport and hydraulics data. The advantage of this

procedure, as was mentioned before, is that measured data are highly congruent because the

transport and hydraulics values are internally linked.

The special tool has three parts: Probe segment (Conductivity and RWT), analog-digital interfacesegment and hand PC where is located the software. The operation of both probes may besimultaneously assuring an efficient inter-calibration. Figure 3 shows left a probe, right the A/Dinterface and at the center the hand PC.


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Figure 3 Special measurement kit for measurements.

The experimental case presented was a measurement journey done in River Cali with a meanwidth= of 25 m, in the city of the same name in Colombia, in 2008. Figure 4. This stream is atypical mountain river of high discharge, high roughness, high slope and high mean velocity of flow. Photos show several aspects of stream at the measurement point.

There were used two tracers: RWT and common salt poured at 423 m. of distance from injection point. RWT is the red line while the salt is the blue one. Screen has a common grid for the two

different measurements. Mass of RWT was 4 g. while mass of common salt was 10178 g.Following it is shown the screens of hand PC once the tracer has passed by the measurement point.

The software has a special filter routine that rejects the high frequency noise viewed as a collectionof spikes on the experimental tracer line, as is shown in photos. The numerical results are displayedalso in a table in screen.


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Figure 4. Some aspects of stream at measurement site

A.- Tracer curves and filter application


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B.- RWT experimental curve & model. Measurement data

C.- Salt experimental curve & model. Measurement data

Figure 5. Several screens with measurements.

Photos of Figure B show experimental RWT curve with its theoretical modelation using a specialroutine that applies Eq. (16) to classical Ficks equation, obtaining a very close simulation of real


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curve. Related data are displayed in table which appears immediately in field. Photos of Figure Cshow experimental salt curve with its theoretical modelation, as in B. Figure.

Analysis of both collections of data shows a mean velocity very near in each experiment (U=0.450and 0.457 m/s), also for transport coefficient (E=2.72 and 2.94 m2/s). There is however a significanterror in discharge (Q=5.76 and 4.32 m3/s). Further analysis in Office indicated that dischargecalculated with salt tracer was more accurate. It is probable that RWT had an error in the massestimation, due the difficulties of measure an accurate volume of this very viscous compound.However, this procedure is useful to get an appropriate calibration of system.

5. CONCLUSIONS.

5.1. - It was explained a new mean flow velocity equation which relates the hydraulics and transportof mass fields. This implies that related parameters are inherently correlated which it is not the casein current approximations.

5.2. - This new approximation also it is based on the waste of energy as heat, as in the classicalChezys equation when the gravitational work is transformed in losses by action of surface friction.In th new case, the Free energy (derived from energy of formation of tracer compound) is convertedentirely in irreversible heat.

5.3. - A study case was presented in a large mountain stream in Colombia, using two tracerssimultaneously (Common salt and RWT). Using the new equations was possible to simulate veryaccurately the tracer real curves; also it was possible to get a congruent table of measured

parameters.

5.4.- The use of two different tracers was useful for an inter calibration procedure, allowing to

detect the error sources in measurement procedures.

AKNOWLEDGEMENTS

Author wish to thank to Amazonas Technologies company who supported the measurements donein Cali city, specially to Mr. Jairo Carvajal.

REFERENCES:

[1] Gribbin J. In search of Schrodingers cat .Bantam Books, New York, 1984.

[2] Martin J.L & McCutcheon S.C. Hydrodynamics and transport for water quality modeling. Lewis,Boca Raton, 1998.


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[3] Jolankay G. Hydrological,chemical and biological processes of contaminat transformation and transport in river and lake systems. UNESCO, Paris, 1992.

[4] Nekrasov B. Hydraulics. Mir. Moscow, 1968.

[5] Vennard J.K. Elementary fluid mechanics. John Wiley & Sons. New York, 1954.[6] Peralta-Fabi R. Fluidos. Fondo de cultura econmica, Mexico,2001.

[7] Kondepudy D. & Prigogine I. Modern thermodynamics John Wiley & Sons, New York, 1998.

[8] Einstein A. Investigations on the theory of Brownian movement. Dover, New York, 1955.

[9] Constain A. & Lemos R. Una ecuacin de velocidad en rgimen no uniforme, su relacin con elfenmeno de dispersin como funcin del tiempo y su aplicacin a los estudios de Calidad de Aguas.

Revista Ingeniera Civil , Diciembre 2011. CEDEX, Madrid.

[10] Constain A. Verification of ergodic principle for a dispersion process in flow. Aqua_Lac, Vol4. No.1. pp19-29. UNESCO, Montevideo.

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