A thermal model for prediction of the Nusselt number in a pipe with chaotic flow
Thierry Lemenand, Hassan Peerhossaini
*Thermofluids and Complex Flows Research Group, Laboratoire de Thermocineetique, CNRS UMR 6607, E
Ecole Polytechnique de l’Universiteede Nantes, Rue Christian Pauc, B.P. 50609, F-44306 Nantes, Cedex 3, France
Received 24December 2001; accepted 3 June 2002
Abstract
It is currently well established that Lagrangian chaos intensifies heat transfer significantly [J. Fluid Mech.
209 (1989) 335]. It thus appears to be a promising technique for the design of compact, high-performance heat exchangers and heat exchanger-reactors. However, the design of such apparatus requires extensive calculations. The objective of this work is to implement a simplified thermal model with which to simulate heat transfer in a twisted pipe (of a shell and tube heat exchanger) of two tube configurations, helically coiled or chaotic, without requiring the heavy calculations needed in the numerical resolution of the Navier–Stokes and energy equations. The large database obtained from the parametric study of the variation of the Nusselt number using the heat transfer model developed here, allows one to correlateNu with Re, Pr, Nbends: Nu¼1:045Re0:303Pr0:287Nbends0:033. This correlation is valid for coil geometry with alter- nating planes of curvature, i.e. chaotic configuration and the range of validity of the correlation is Re2 ½100;300,Pr2 ½30;100andNbends2 ½3;13.
Ó 2002 Elsevier Science Ltd. All rights reserved.
Keywords:Chaos; Chaotic advection; Conservative dynamical system; Convective heat transfer; Heat exchanger; Heat exchanger-reactor; Numerical modelling; Nusselt number
1. Introduction
Chaotic advection in a twisted-pipe flow has been the subject of very active research in the past decade [1–12]. In a heat exchanger-reactor with chaotic advection effect, the fluid particles travel
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E-mail addresses: [email protected] (T. Lemenand), [email protected] nantes.fr (H. Peerhossaini).
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close to the hot zones (i.e. in the vicinity of the tube wall) as well as in the cold zones (i.e. in the flow core). On the one hand, recent experimental studies [3,5,7,8,11], and numerical studies [6,9,10] have shown that the phenomenon of chaotic advection considerably increases heat transfer. On the other hand, the results of Mokrani et al. [8] reveal that the presence of chaotic trajectories permit more homogeneous temperature distributions within the fluid. The obvious practical interest of these chaotic flows warrants their study in order to develop tools and guidelines for the design of chaotic heat exchangers.
Nomenclature
Cp specific heat at constant pressure, J kg1K1 d pipe diameter, m
De Dean number¼Reðd=RÞ0:5 N number of injected particles Nbends number of bends
Nu Nusselt number
p power unit for one particle, W Pr Prandtl number
_ q
q000 heat flux density, W m2 r pipe radius, m
R radius of curvature of the pipe, m Re Reynolds number
S heat transfer surface area, m2
t time, s
T temperature, K V axial velocity, m s1 x, y local coordinates, m
y0 particles-injection distance, m z curvilinear coordinate, m Greeks
D increase
k thermal conductivity, W m1K1 q density, kg m3
h angular abscissa, rad Subscripts
crown in the crown
exchange wall of the whole heat exchanger
m mean
0 at the surface
section in the heat exchanger section total in the whole heat exchanger
The basic idea of chaotic advection is rather simple. If a fluid particle is so light, so inert and so passive that it follows the fluid in which it is immersed everywhere and at any moment, its equations of motion are simply the equations of advection:
d~xx
dt ¼Vðx;!tÞ
ð1Þ where~VV is the velocity field of the fluid, considered known or obtained by other means.
From the point of view of dynamical system theory, the format of the problem presented by Eq.
(1)––i.e.n coupled differential equations in a space of dimensionn––is sufficiently rich for nP2 that in general one must suppose that the solutions are technically chaotic except under rather restricted conditions. Therefore obtaining chaotic solutions for system (1) is not a surprise: similar cases are found in other fields of physics, for example the movement of charged particles subjected to electric and magnetic fields. However, the major difficulty in the chaotic field of advection lies in the complexity of obtaining the basic flow field rate for configurations of current technological interest. The alternating Dean flow at the base of the chaotic design of heat exchanger is an example. Therefore the search for a simplified model for heat exchanger design requires as a first step a solution for the basic flow.
The primary objective of the present work is to study the effect of Lagrangian chaos on the Nusselt number in a twisted pipe and consequently on the heat transfer in shell and tube heat exchangers with tubes arranged in chaotic configuration. The geometries considered are those of Mokrani et al. [8]. The solution of the basic flow adopted here is that of Dean [13] for curved channels, expressed in the form of a series in powers of a disturbance (asymptotic expansion), where De is the Dean number defined as De¼Reðd=RÞ0:5. This solution is limited to Dean numbers lower than 55. It has been used by Jones et al. [1] to develop a simple model for cal- culating the fluid trajectories in the flow inside a helical tube with alternating planes of curvature, which is likely to generate chaotic particle trajectories.
2. Alternating Dean flow model
The flow through a curved channel or pipe is called Dean flow. In such a configuration, cen- trifugal forces generate a secondary flow that consists of two counterrotating cells, generally called Dean roll-cells. Jones et al. [1] have shown that fluid particle trajectories are not chaotic in a laminar helical flow but that they may become chaotic if a geometrical perturbation is added to the flow. Such a perturbation can be achieved merely by periodically shifting the curvature plane by90°. The resulting geometry is called ‘‘alternating Dean flow’’. It is important to notice that chaotic trajectories can appear at very low-Reynolds numbers, i.e. in a laminar regime (from a Eulerian point of view). Dean vortices are generated due to the imbalance between the centrifugal force and the radial pressure gradient in the tube cross-section, and this non-equilibrium exists independent from the Reynolds number. Therefore, there is no critical value of the Dean number for the appearance of the vortices and generation of chaotic trajectories.
The present study is primarily numerical. Fluid particle trajectories in alternating Dean flow are calculated by using the model proposed by Jones et al. [1]. This model uses the velocity field in a curved channel of circular section, approximated by DeanÕs asymptotic solution [13,14]. The
hypotheses of a steady, laminar and incompressible flow are admitted. In this model, the equa- tions of particle trajectories are given [1] by the following system of equations, written in the local coordinates (x;y;h) of Fig. 1:
dx dh¼ Re
144ð45x223y2þ8x2y2þx4þ7y4Þ dy
dh¼Re
24xyð3x2y2Þ 8>
<
>: ð2Þ
where dx=dhand dy=dhrepresent the velocities in thexandydirections, non-dimensionalised by the flow mean velocity.
In a helical tube, the trajectory of a particle can be calculated by integrating a system of non- linear equations between the particleÕs initial position and the angular coordinate of its final position. In an alternating Dean flow, the plane of curvature of each bend is shifted by90°from that of the preceding bend (see Fig. 1). To calculate the trajectories of the particles in this flow, a change of frame of reference is used at the exit of each bend in order to follow the positions of the particles in the next bend with the same equations. One then positions the particles at the entry of the next bend. After integration of the system, the solution is used as input conditions for the following bend. The system of differential equations was integrated by the Runge–Kutta method of the fourth order with adaptive steps.
It should be emphasized that the model presented here is limited by the basic assumptions used in its construction, in particular the validity of the approximate solution of Dean for small Dean numbers and the instantaneous modification of the Dean roll-cells at the entry of each bend.
However, the simplicity of this model does not prevent it from capturing the essential charac- teristics of Lagrangian chaos or from distinguishing these characteristics from those of a regular flow. Moreover, this simplicity allows one to obtain the particle trajectories, and especially the extent of chaos of these trajectories, in computations taking only a few hours or even less on a workstation, whereas the same type of information requires much longer calculations if one solves the complete system of Navier–Stokes and energy equations.
3. Heat transfer model
The hydrodynamic representation of the flows by a dynamical system such as that above lets one treat convective heat transfer with the same approach by using the assumption that heat is transported like a passive tracer by the flow. Thus heat transfer from the walls of the heat ex- changer towards the fluid (to be heated) is transformed, by analogy, to the transport of a pop-
Fig. 1. Local coordinates associated with coiled tube.
ulation of a passive tracer from the wall towards the core of the flow. However, unlike the heat transferred to the fluid at the wall, the tracer particles cannot be introduced directly at the wall because of the no-slip condition, which would prevent the particles from leaving their initial position. To obviate this obstacle, particles in this model are introduced into the flow at a distance y0 from the wall (10% of the radius), a distance determined by comparison of the model results with experimental results in a helical flow.
The main purpose here is to develop a reduced model that can simulate heat transfer in al- ternating Dean flow without solving the complete Navier–Stokes and energy equations. This simplification is achieved by assuming, on the one hand, that heat or mass transfer by molecular diffusion is negligible compared to transfer by advection, and on the other that natural convection is negligible compared to forced convection. In other words, this is a purely advective and con- stant-fluid-property model.
The basic piece of this simplified thermal model is the expression of wall/fluid transfer phe- nomena by the continuous injection from the wall of passive particles. This mimics a constant- heat-flux condition and the particle density then represents the flux density: each particle represents a power density. In this work 1100 particles are injected per bend. They are distributed regularly on the wall (i.e. 20 particles per section and 55 sections per bend) in order to obtain the same curvilinear distance (longitudinal or transverse) between two neighbouring particles. Tests were carried out in which 10 times more particles were injected, but the Nusselt numbers remained unchanged. Let us define the powerp represented by a particle as
p¼qq_000Sexchange
Ntotal ð3Þ
whereqq_000 is the wall heat flux and Sexchange is the wall surface area.
The average temperature is deduced by carrying out an energy balance on a differential cy- lindrical control volume of the tube of thickness dz:
pNsection¼qðpr2dzÞCpDTm
dt ð4Þ
whereDTm is the mean temperature increase in the control volume. Hence
DTm¼ pNsection
qpr2CpVm
ð5Þ
The surface temperature cannot be directly calculated from Eq. (5) as the average temperature, since the particles were not injected directly at the wall but at a distance y0 from it. One thus calculates the temperature at this distancey0 by making an energy balance as previously but this time on a crown centered aroundy0:
DTy0 ¼ pNcrown qScrownVy0Cp
ð6Þ
The wall surface temperature T0 can then be calculated by making an additional assumption that considers a conduction layer between the wall and the radial positiony0. This assumption is justified by the no-slip condition on the wall and the very small effect of advection in a zone very
close to the wall. Therefore, one obtains a linear evolution of the temperature (FourierÕs law) between the wall and the position y0:
T0¼Ty0 þqy0
k ð7Þ
From the above relations, one can deduce the different thermal characteristics of the flow, including the Nusselt number, for the two possible configurations: helically coiled and chaotic.
4. Results and discussion
Calculations were carried out for a given geometry (r=R¼0:091) and fluid characteristics of water except for the viscosity, which was between 4.5 and 15 times greater than that of water, thus the Prandtl number varies from 30 to 100. The choice of Prandtl numbers greater than that of water was guided by the industrial applications of this type of heat exchanger. Calculations were then carried out for different Reynolds numbers (from 100 to 300), different numbers of bends (from 3 to 13) and a constant wall heat flux qq_000 ¼2050 W m2.
The objective of this work was to obtain design correlations from the computation of a purely advective (non-diffusive) flow field and its corresponding thermal model. For this purpose a ge- neric algorithm [15] was used to explore the correlation between the Nusselt number and other problem parameters. It uses the principle of the survival of the most adapted structures and pseudo-random information exchange to construct a search algorithm that has certain charac- teristics of human search. Differing from conventional search techniques, the common feature of the genetic algorithms is to simulate the search process of natural evolution and take advantage of the Darwinian survival of the fittest principle. In short, evolutionary algorithms start with an arbitrarily initialised population of coded individuals, each of which represents a search point in the space of potential solution. The goodness of each individual is evaluated by a fitness function which is defined from the objective function of the optimisation problem. Then, the population evolves toward increasingly better regions of the search space by means of both random and probabilistic biological operations. The basics operators used in genetic algorithms consist of selection (the selection of parents for breeding), cross-over (the exchange of parental information to create children) and mutation (the changing of an individual). Note however here that the ergodicity of the biological operators makes them potentially effective at performing global search (in probability). Also, genetic algorithms have the attribute of a probabilistic evolutionary search and are neither bound to assumptions regarding continuity nor limited by required prerequisites.
4.1. The experimental correlations and the present model for helically coiled tube
As previously explained, the particle injection is done at a distancey0 from the wall that is not determined by the model; it is the model free parameter and is determined by fitting model results to experimental results. Janssen and Hoodendoorn [16] provide an empirical correlation giving the Nusselt number Nu (with an accuracy of20%) in a helically coiled heat exchanger tube in forced convection with constant wall heat flux:
Nu¼0:32 þ3r
R
Re0:5Pr0:33 2r z
0:14þ0:8ðr=RÞ
ð8Þ The experimental correlation (8) is valid for Dean numbersDebetween 20 and 830 (that is to say, in this study, Reynolds numbers between 66 and 2750), Prandtl numbers between 30 and 450, and geometrical ratiosr/Rbetween 0.01 and 0.08.
The first task is to find the particle-injection positiony0 for each Reynolds number so that the model, applied to the helical configuration, correctly predicts Nusselt numbers close to those given by the experimental correlation (8). When the various particle-injection positions are known for the helical configuration, they are used in simulation of the flow and the heat transfer coefficient in the chaotic configuration.
Fig. 2 shows that the model predicts a correct Nusselt number evolution judged by the ex- perimental correlation (8) and is included in the20% accuracy interval of this correlation. For Reynolds numbers greater than 100, the model predicts Nusselt numbers lower than the corre- lation but always within the20% interval, and this remains true up to Reynolds number 300.
For Reynolds numbers above 300, however, the model predicts Nusselt number more than 20%
lower than the experimental correlation, thus limiting the validity of the model to Reynolds numbers below 300. Similarly, for Reynolds numbers less than 100, the deviation from the ex- perimental correlation is again more than 20%. Nevertheless, it should be recalled that, on the one hand, the experimental correlation (8) is valid for Reynolds numbers greater than 66 (in the present geometry) and, on the other, the hydrodynamic model itself is no longer valid for high Reynolds numbers (the Dean solution is valid only for moderate Reynolds numbers).
The computation was repeated for Reynolds numbers between 150 and 300 and for Prandtl numbers between 30 and 100, from which the injection distance y0 was deduced. The generic algorithm was used to correlatey0 withRe and Pr. It yields
y0¼183:0Re0:298Pr0:306 ð9Þ
Fig. 2. Nusselt number versus number of bends (helical configuration).
with a confidence interval of 10%. This correlation shows that the injector position varies in- versely with the Reynolds and Prandtl numbers. For the range ofRe andPr considered here, y0
varies between 8% and 15% of the tube inner radius.
4.2. The model predictions for the helical and chaotic configuration
One notes first in Fig. 3 that the higher the Reynolds number, the higher the Nusselt number predicted by the model in both configurations. This corresponds to the well-known fact that stirring increases with fluid velocity. Secondly, it is easily discerned that the model predicts higher Nusselt numbers in the chaotic configuration than in the helical configuration. This is in complete agreement with [5–8], which found higher heat-exchanger efficiencies in the corresponding ex- periments. Moreover, the Nusselt number shows a constant evolution according to the number of bends in the chaotic configuration (for higher Reynolds numbers), while it decreases continuously in the helically coiled configuration; this phenomenon is discussed in Section 4.3.1 below.
4.3. Correlation
The fluid advection and heat transfer coefficient were computed in the chaotic heat exchanger for the various ranges of parameters described previously by solving the system of Eq. (2) and by using the thermal model developed here. An example of particle distribution in the pipe-section for the chaotic and also non-chaotic configurations found by solving the system of Eq. (2) is given in Fig. 4. It can readily be seen that the chaotic twisted-pipe flow provides better stirring and hence mixing of the passive tracer, as found in [1]. Therefore, the Nusselt number is expected to be higher in the chaotic case than in the helically coiled case. It should be noted that the particle distributions in Fig. 4are obtained after only 13 bends (i.e. four rings and one bend); therefore, the chaos generated is not fully developed and some traces of the regular Dean roll cells still exist.
Fig. 3. Nusselt number versus number of bends (helical and chaotic configurations).
A larger number of bends will show a more homogeneous distribution of the tracer particles in the pipe cross-section. However, since the objective of this work was to investigate the effect of a small number of axis rotations on chaos generation, we have been content with the partial chaos ob- tained within only four rings. A criterion for the chaos generated in a twisted pipe of a limited number of bends was developed by Peerhossaini and coworkers [9].
4.3.1. Influence of the number of bends
Fig. 3 shows that the number of bends does not have a significant influence on the Nusselt number in the chaotic configuration. For most of Reynolds numbers tested here (except for 100 and 150), the Nusselt number does not evolve beyond seven bends. Recent experiments [17] on mixing of a fluorescent tracer with water at comparable Reynolds numbers show that an almost homogeneous mixing is obtained after six bends. This almost stable evolution comes from the destruction and reorganization of the flow structure (Dean vortices) at each bend rotation. In- deed, in the helical case, one observes a continuous thickening of the boundary layer with the progression downstream in the heat exchanger, which decreases heat transfer to the wall and thus the Nusselt number. In the chaotic case, the rotation of the 90°bends breaks down the Dean cells, each time reducing the boundary layer thickness, and makes it possible to maintain the heat
Fig. 4. Particle distribution at exit of the 13th bend (Pr¼40).
transfer, and thus the Nusselt number, at a constant level. This behavior is completely different from that in the helical case where the Nusselt number decreases continuously in the downstream direction; correlation (8) gives its evolution in thez direction as z0;26.
In the chaotic configuration, the apparent increase of the Nusselt number with the number of bends at low-Reynolds numbers (Re<150) can be attributed to the developing nature of the Dean roll-cells. In fact when the flow Reynolds number or the angular extent of the bends is insufficient to generate fully developed Dean roll-cells, the vortices continue their development in the downstream bends. This development is accompanied by augmentation of mixing and heat transfer in the downstream direction, as shown in Fig. 3.
At higher Prandtl numbers, however, the chaotic configuration shows a very slight decreasing tendency for the Nusselt number according to the number of bends (Fig. 6) that can be ap- proximated by a power law Nbends0:033. Nevertheless, this decrease with z is an order of magnitude smaller than what is observed experimentally and reflected in the correlation (8) for the helically coiled tubes.
Fig. 5 shows the enhancement of the Nusselt number in the chaotic configuration over that in the helical coil. It increases with the number of bends, largely because of the decrease in the Nusselt number of the helically coiled tube with bend number. However, the latter seems to approach an asymptotic value and this increase should saturate at high bend numbers (Fig. 6).
4.3.2. Influence of Reynolds number
On Fig. 7 it is clear that the Nusselt number grows with the Reynolds number, showing the same tendency as observed in the helical configuration except that the evolution of the Nusselt number is less sensitive to Reynolds number variation in the chaotic than in the helical config- uration. At high Reynolds numbers (Re>200), the augmentation of the Nusselt number withRe is independent of the number of bends. The experimental correlation (8) provides an evolution in Re0:5 for helically coiled tubes, whereas one finds an evolution in Re0:3 for the chaotic configu-
Fig. 5. Enhancement of Nusselt number by chaotic advection over the Nusselt number in helically coiled tube.
ration. This can be explained by the fact that in the helical configuration, it is the dynamics, i.e.
the strength of the Dean vortices that has a dominant role in the mixing, whereas in the chaotic configuration the mixing in the cross-section depends little on the intensity of the secondary flow, since the chaotic behavior of the trajectories is inherent in the flow kinematics [8].
4.3.3. Correlation
The parametric study of the Nusselt number presented above shows that it varies with three main design parameters,Re, Prand Nbends. Therefore, from the numerical results of the thermal
Fig. 6. Nusselt number versus number of bends (chaotic configuration).
Fig. 7. Nusselt number versus Reynolds number (chaotic configuration).
model (Section 3), the following correlation was established by using the genetic algorithm de- scribed in Section 4in the case of the chaotic configuration:
Nu¼1:045Re0:303Pr0:287Nbends0:033 ð10Þ
This correlation fits the results obtained from the model with a confidence interval of25% and standard deviation of 7%. The range of validity of the correlation isRe2 ½100;300,Pr2 ½30;100 and Nbends 2 ½3;13. It should be noted that the maximum deviation of the correlation from the model results occurs forRe¼100. The confidence interval is 18% for Reynolds number between 150 and 300. Fig. 8 shows the Nusselt number correlation data plotted versus the Prandtl number
Fig. 8. Nusselt number correlation for chaotic coil as a function of Prandtl number.
Fig. 9. Nusselt number correlation for chaotic coil as a function of Reynolds number.
for various parameter values,Refrom 100 to 300 and bend numbers from 3 to 13. Fig. 9 shows the same correlation data plotted against the Reynolds number forPrfrom 30 to 100.
5. Conclusions
Chaotic advection of passive fluid elements in a twisted pipe has been well documented since the pioneer work of Jones et al. [1] on a simplified low-Reynolds-number model. Later experimental work on the fundamental aspects [3,4,9,12], heat-exchanger efficiency [5–8,11] and mass transfer effectiveness [10] of chaotic advection demonstrated the great technological interest of this special flow, and warranted the search for a simplified heat transfer model analogous to the hydrody- namic model [1].
The present study proposes a heat transfer model analogous to the simplified hydrodynamic model for the prediction of the Nusselt number for a chaotic twisted pipe with constant wall heat flux. The chaotic twisted pipe is composed of a succession of quarter-circle bent tubes with 90°
angle of rotation of the curvature plane.
The large database obtained from the parametric study of the variation of the Nusselt number using the heat transfer model developed here allows one to correlateNuwithRe,Pr,Nbends. The correlation (10), valid for chaotic heat transfer in twisted pipes, shows a weaker dependence ofNu onReandPrthan the experimental correlation (8) obtained for a helically coiled tube. In different steps of development of the correlation (10), a physical interpretation of the behavior ofNuwith flow parameters is given. To obtain a more general correlation than (10), further work should focus on the variation ofNuwith r=R.
Acknowledgements
The authors are grateful to the Centre National de la Recherche Scientifique (CNRS) for funding this project in the framework of ECODEV program of the ARC CNRS-ADEME
‘‘EEchangeurs thermiques’’. The contributions of Dr. A. Mokrani at the early stages of this work and Dr. O. Lottin for making available the genetic algorithm are greatly appreciated.
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