NUMERICAL ANALYSIS OF NATURAL CONVECTION INA CYLINDRICAL ENVELOPE WITH AN INTERNALCONCENTRIC CYLINDER WITH SLOTS

Kun Zhang1, Mo Yang1, and Yuwen Zhang21College of Power Engineering, University of Shanghai for Science andTechnology, Shanghai, People’s Republic of China2Department of Mechanical and Aerospace Engineering, University ofMissouri, Columbia, Missouri, USA

Detailed numerical analysis is presented for natural convection heat transfer in a cylindrical

envelope with an internal concentric cylinder with slots. Governing equations are discretized

using the finite volume method, and solved using SIMPLE algorithm with QUICK scheme.

The results show that the system can reach steady state and be symmetric when the

Rayleigh number is below 4� 105. When the Rayleigh number is greater than 6� 105, an

asymmetric periodical solution is obtained although the initial field and boundary conditions

were symmetric. As the Rayleigh numbers increase further, a quasi-periodic solution of

the system is achieved at Ra¼ 2� 106, and the periodicity is lost at Ra¼ 6� 106. It is

ascertained that the oscillatory flow undergoes several bifurcations and ultimately evolves

to a chaotic flow.

1. INTRODUCTION

The present work is motivated by natural convection heat transfer in anenclosed isolated-phase busbar used for transmitting large electric current in powerplants. The busbar consists of two metal cylinders: an inner hollow cylinder (currentbusbar: the cross-section can be circular, hexagon, or octagon) and an outer metalcylinder. The heat generated in the busbar from the Joule heating is transferred tothe outer envelope by radiation and natural convection. The flow and thermal fieldsare well studied for the natural convection induced by internal heating in horizontalconcentric cylindrical annuli. Powe et al. [1, 2] visualized the flow patterns with smokeas a tracer and obtained the Rayleigh number at which the flow will change from asteady flow to an unsteady flow. Rao et al. [3] investigated the transient oscillatoryphenomena and the numerical determination of the critical Rayleigh number atwhich unsteady flow occurs. Cheddadi et al. [4] studied the bifurcation phenomenon

Received 28 July 2010; accepted 23 February 2011.

Financial support from the Chinese National Natural Science Foundation under grants nos.

50876067 and 50828601, Project of Shanghai Education Committee 10ZZ91, and Shanghai Leading

Academic Discipline Project J50501 are gratefully acknowledged.

Address correspondence to Mo Yang, College of Power Engineering, University of Shanghai for

Science and Technology, Shanghai 200093, People’s Republic of China. E-mail: yangm66@gmail.com

Numerical Heat Transfer, Part A, 59: 739–754, 2011

Copyright # Taylor & Francis Group, LLC

ISSN: 1040-7782 print=1521-0634 online

DOI: 10.1080/10407782.2011.572762

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of natural convection at different initial conditions, and showed that flow pattern wasnot unique and depended on the initial conditions at a high Rayleigh number. Liuet al. [5] investigated the stability of natural convection by measuring the total heattransfer coefficient and the distribution of radiation temperature of water, air, andsilicone. The results indicated that the critical Rayleigh number dictates the transitionfrom steady to unsteady. Mizushima et al. [6] numerically investigated the bifurcationphenomenon and obtained the critical Rayleigh number when flows changing intomulti-vortex. Yoo [7] investigated numerically, the bifurcation sequences to the chaosfor the natural convection in horizontal concentric annuli in detail.

Comparatively, few works have been reported on unsteady natural convectionheat transfer in a more complex domain, such as in a cylindrical envelope with aninternal concentric cylinder with slots that will be studied here. Kuleek assumed thatthe heat transfer enhancement of a cylindrical envelope with an internal slotted cyl-inder was expected to be about 30–40% more than that of the concentric cylindricalannuli [8]. Wang et al. [9] experimentally investigated natural convection in anenclosed isolated-phase electric current bus, and found that the convective heattransfer coefficient with slots could be enhanced by as much as 50%. Yang et al.[10] numerically investigated flow and heat transfer in steady and symmetric domain.Li et al. [11] experimentally studied three different heat transfer regimes under differ-ent Rayleigh numbers. Zhang et al. [12] experimentally investigated steady-statenatural convection heat transfer between a cylindrical envelope and an internal con-centric heated octagonal cylinder with and without slots. Yang et al. [13] performeda numerical solution of the same problem and their results were compared withexperimental results by Zhang et al. [12]. It was reported that the maximum relativedifference between the computed average equivalent thermal conductivities andexperimental values by reference [13] was less than 9%. The existing experimental

NOMENCLATURE

a thermal diffusivity, m2=s

F dimensionless time

g gravitational acceleration, m=s2

Keq average dimensionless equivalent thermal

conductivity of whole cylinder

K relative thermal conductivity

ks thermal conductivity for solid, W=mk

kf thermal conductivity for fluid, W=mk

L gap width, m

p pressure, Pa

P dimensionless pressure

Pr Prandtl number

Q heat transfer rate of whole circle, W

ri radius of slotted inner circle, m

ro radius of envelope circle, m

R dimensionless radial coordinate

Ra Rayleigh number

S relative slot width

Ti temperature of slotted inner circle, �C

To temperature of envelope circle, �Cu tangential velocity, m=s

U dimensionless tangential velocity

UR characteristic velocity, m=s

V dimensionless radial velocity

a thermal diffusivity

b coefficient of thermal expansion, 1=�CC nominal diffusion coefficients in the

momentum equations

CT nominal diffusion coefficients in energy

equation

d thickness of slotted inner circle, m

h angular coordinate

H dimensionless temperature

q density, kg=m3

s time, s

n kinematic viscosity, m2=s

/ angle

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investigation of natural convection in a cylindrical envelope with an internal concen-tric cylinder with slots has focused on steady and symmetrical flow fields.

Natural convection heat transfer is a recurrent phenomenon in the world andmost of these flows are unsteady, especially those encountered in engineering appli-cations. Studies of unsteady-state natural convections have attracted increasinginterest over the last few decades due to a desire to improve the phenomenologicalunderstanding of natural convection, and the pressing need for numerical modelscapable of predicting the corresponding flow structures and related heat transfer[14]. Many helpful results from the research of natural convection indicated that alaminar steady flow and heat transfer is observed for lower Rayleigh number, whileunsteady convection is obtained at high Rayleigh numbers [15–17]. Victorri andBlondeaux [18] numerically studied the transition process that leads to the oscillatoryflow over a wave from a periodic behavior to chaos using Feigenbaum scenario.They found that by increasing the Reynolds number, the flow experienced an infinitebifurcation sequence of period doubling that took place at successive critical values.Numerical studies of natural convection in a small aspect ratio enclosure were per-formed by Mokutmoni and Yang [19, 20], and their results verified the conclusionsof Victorri [18] that the flow becomes turbulent though diploid periodical bifurca-tion. The first transition was a bifurcation from steady to periodic oscillatory flow,and the second transition is a sub-harmonic bifurcation as the Rayleigh numberincreased further. The bifurcation sequence to temporal chaos has been an importantsubject in many areas of nonlinear systems, and has also been studied widely by thefluid mechanics community. Recently, a great deal of literature relevant to naturalconvection has concentrated on the transition process to unsteady periodic flowand route to chaos. Deshpanda and Srinidhi [21] studied the mixed convection ina rectangular parallelepiped and observed certain features of dynamic systems likebifurcation, period doubling, and chaos. Aklouche [14] studied the unsteady naturalconvection in an air-filled square enclosure. They plotted the temporal evolution ofthe hot global Nusselt number and the attractors in a space trajectory, and discussedthe effect of the Rayleigh number on the route to chaos.

The objective of this article is to numerically simulate natural convection heattransfer in a cylindrical envelope with an internal concentric cylinder with slots. Theeffects of Rayleigh numbers on route to chaos and the study of the periodic, thequasi-periodic, and the chaotic regimes will be investigated in detail. The governingequations will be discretized using the finite volume method based on staggered gridformulation, and solved using the SIMPLE algorithm [22] with QUICK scheme.Special attention will be paid to the treatment of the internal slotted concentric cyl-inder, which can be regarded as two isolated islands, and an integrative solutionmethod [23] will be used to solve the problem of isolated islands. The space trajectoryof velocity at the sample point and the temporal evolution of the average dimension-less equivalent thermal conductivity (Keq) will be studied to investigate the effect ofthe increasing Rayleigh numbers on the routes to chaos.

2. PROBLEM FORMULATIONS

A schematic diagram of the physical model under consideration is shownin Figure 1. The inner and outer cylinders are kept at uniform except for different

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temperatures Ti and To, respectively, with Ti>To. As a result of the temperature dif-ference between the two circles, density gradients occur and lead to natural convec-tion. It is assumed that the fluid in the enclosure is of a Boussinesq type, and the fluidflow and heat transfer is two-dimensional and laminar.

The dimensionless governing equations are

qUqF

þ VqUqR

þU

R

qUqh

¼ � 1

R

qPqh

þ C1

R

qqR

RqUqR

� �þ 1

R2

q2U

qh2

" #þ Sh ð1Þ

qVqF

þ VqVqR

þU

R

qVqh

¼ � qPqR

þ 1

R

qqR

RqVqR

� �þ 1

R2

q2V

qh2

" #þ SR ð2Þ

qHqF

þ VqHqR

þU

R

qHqh

¼ CT1

R

qqR

RqHqR

� �þ 1

R2

q2H

qh2

" #ð3Þ

qVqR

þ V

Rþ 1

R

qUqh

¼ 0 ð4Þ

where the source terms are

Sh ¼ �UV

Rþ Cu � U

R2þ 2

R2

qVqh

� ��H sin h ð5Þ

SR ¼ �U2

Rþ Cv � V

R2þ 2

R2

qUqh

� �þH cos h ð6Þ

Figure 1. Physical model.

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The dimensionless parameters used in Eqs. (1)–(6) are defined as follows.

R ¼ r

L; F ¼ UR

sL; H ¼ T � To

Ti � To; P ¼ p

qU2R

; U ¼ uhUR

; V ¼ vrUR

; K ¼ k

kf

ð7Þ

The dimensionless parameters Rayleigh number Ra, Prandtl number Pr, and refer-enced velocity UR, are defined as

Ra ¼ bgL3ðTi � ToÞan

; Pr ¼ na; UR ¼ ðRaPrÞ1=2 a

Lð8Þ

where b, a, and n are thermal expansion coefficient, thermal diffusivity and kinema-tical viscosity of the fluid, respectively. L is the gap width of annulus, and g is thegravitational acceleration.

The nominal diffusion coefficients C and CT in the momentum and energyequations are, respectively, defined as

C ¼ Pr

ðRaPrÞ1=2; CT ¼ K

1

ðRaPrÞ1=2ð9Þ

where K¼ k=kf is the relative thermal conductivity, which takes the value of 1 in thefluid region and ks=kf in the solid region.

The boundary conditions for Eqs. (1)–(4) are as follows.

R ¼ Ri; U ¼ V ¼ 0; H ¼ 1 ð10Þ

R ¼ Ro; U ¼ V ¼ 0; H ¼ 0 ð11ÞThe following periodical boundary conditions are given at h¼ 0 and h¼ 2p.

Uðh ¼ 0;RÞ ¼ Uðh ¼ 2p;RÞ ð12Þ

Vðh ¼ 0;RÞ ¼ Vðh ¼ 2p;RÞ ð13Þ

Hðh ¼ 0;RÞ ¼ Hðh ¼ 2p;RÞ ð14Þ

Apart from the above boundary conditions, the following two additionalconditions for the internal solid cylinder must be satisfied.

Ri � d L � R � Ri; / � h � p� /; U ¼ V ¼ 0; H ¼ 1 ð15Þ

The initial condition is

F ¼ 0; U ¼ V ¼ H ¼ 0 ð16Þ

To observe the total heat transfer effect, the average dimensionless equivalentthermal conductivity based on the whole outer circle is defined as

Keq ¼Q

2pkf ðTi � ToÞlnrori

ð17Þ

where Q is the total heat transfer rate.

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The slot width is defined as

S ¼ 2a=p ð18Þ

3. NUMERICAL PROCEDURES

The governing equations were discretized by the finite volume method, and thecentral difference scheme was adopted for the discretization of the convection-diffusion terms to decrease the numerical viscosity. The SIMPLE algorithm withQuick scheme was used for handling the pressure velocity coupling. Since in theinterior of the flow field there is an internal slotted concentric cylinder, which canbe regarded as two isolated solid regions, this is a conjugated-type heat transferproblem. The whole annulus is taken as the solution domain, and the temperaturesand velocities are determined simultaneously for both fluid and solid regions. Anintegrative solution method is adopted, in which both fluid and solid regions are gov-erned by the same partial differential equations. The velocities at the grid point in thesolid regions should always be zero, and the temperatures of the internal slotted cyl-inder are always 1. The algebraic equations resulting from the control volumeapproach are written as

aPUP ¼X

anbUnb þ b ð19Þ

To set the desired value of UP equal to B, one can let ap¼A and b¼A �B in Eq. (19),where A is a very large number. The value of UP obtained from Eq. (19) becomes

UP ¼�X

anbUnb=A

�þ A � ðB=AÞ ¼ B ð20Þ

This method can also be applied in the internal region to the momentum equa-tion to obtain zero velocity by setting B¼ 0.

The parameters used in this study are: ro=L¼ 1.625, ri=L¼ 0.625, d=L¼ 0.1125,Pr¼ 0.7, and S¼ 0.1.

4. RESULTS AND DISCUSSION

4.1. Steady-State Solutions

The grid-independence of the numerical results is studied for the case ofRa¼ 105. When the time step is equal to 0.01, the six mesh sizes in Table 1 are usedto complete the grid-independence study. It is noted that the total numbers of gridpoints for the above six mesh size are 80� 40, 100� 50, 120� 50, 100� 70,

Table 1. Equivalent thermal conductivities for different grid numbers

Grid 80� 40 100� 50 120� 50 100� 70 150� 80 200� 100

Keq 5.1184 5.0579 5.0375 4.9931 4.9371 4.8894

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150� 80, and 200� 100, respectively. Numerical experiments showed that the rela-tive error of Keq corresponding to a grid number of 100� 50 and 200� 100 is lessthan 3.5%; 100� 50 grid points are adequate to yield accurate results. Table 2 showsthat the Keq of different time steps for 100� 50 grid points. Seen in this table, thedifference between Keq obtained using the time steps of 0.01 and 0.001 is negligible.So the time step is selected as 0.01 for the present computations. In order to get highresolution for the slotted place, the grid lines are more closely packed in the slottedplace and an internal concentric slotted cylinder. Figure 2 shows the grid system usedin this study.

Average dimensionless equivalent thermal conductivity Keq at differentRayleigh numbers are calculated numerically. The computational results inFigure 3 indicate that the numerical results agreed very well with those obtainedexperimentally by Wang et al. [9]. The maximum relative difference in Keq is less than10%. It is worth mentioning that for Ra¼ 106, the value of thermal conductivity Keq

oscillate with time in our calculation. The numerical result of Keq is thetime-averaged value in Figure 3. The change of thermal conductivity Keq with timehas not been discussed by Wang et al. [9]. However, the authors believe that forRa¼ 106 the experimental result is also the time-averaged value.

For the case of Ra� 4� 105, the numerical simulation revealed that the steadysymmetrical velocity and temperature field can be reached from the initial zero velo-city and uniform field. The streamlines and temperature field for the case ofRa¼ 4� 105 were shown in Figure 4. The streamlines and the temperature field

Table 2. Equivalent thermal conductivities for different time steps

Time step 0.1 0.05 0.01 0.005 0.001

Keq 5.0581 5.0581 5.0579 5.0579 5.05771

Figure 2. Sample of computational mesh.

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are in good agreement with the steady numerical solution with axisymmetric bound-ary condition. Both steady and unsteady numerical solutions have the same resultsfor cases with low Rayleigh numbers.

4.2. Periodic Asymmetrical Solution

When the Rayleigh numbers are greater than 4.5� 105, a periodic time-dependent behavior is observed. Attention is especially paid to the appearance ofthe periodic oscillation, which is observed in a narrow band of Rayleigh numbers

Figure 3. Comparison of numerical and experimental results of Keq.

Figure 4. Steady-state results for Ra¼ 4� 105. (a) Streamline and (b) Isotherms.

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from 4.5� 105 to 1.9� 106. The periodic oscillation phenomenon of numericalsolution for this scope is now analyzed.

A simulation of the natural convection from the zero initial velocity and uni-form temperature fields is performed for Ra¼ 106. When time infinities are suffi-ciently long, a periodic solution is obtained as shown in Figures 5 and 6. Thetemperature and flow pattern change at different steps of a cycle. There are fivebig vortices in the whole cylindrical envelope: four outside and one inside the slottedcylinder. The top vortex on the left moves down along the annulus from the begin-ning of a cycle. Almost at the same time, the top vortex on the right moves up. Thefollowing flow pattern is opposite, and the process is repeated again and again,leading to the flow oscillation. The temperature field is also altered periodically,as depicted by Figure 6. It should be noted that the oscillation of the Keq areclosely related to these flow pattern oscillations. Since vortex moves up with the

Figure 5. Periodic evolution of flow field in a cycle at Ra¼ 106. (a) F¼ 2,001.62 or F¼ 2,017.44, (b)

F¼ 2,005.93, (c) F¼ 2,009.53, and (d) F¼ 2,013.84.

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corresponding one moving down, occuring alternately at the left and right, the ther-mal conductivity have an approximate phase in their oscillation curves. The mini-mum values of the Keq appear at dimensionless time F¼ 2,001.62 and F¼ 2,009.53respectively, and the maximums are at F¼ 2,005.93 and F¼ 2,013.84. The streamlineand isotherm are of single periodicity, and the oscillating period is 15.82, while theaverage thermal conductivity is of a bi-periodic type.

For the case of Ra¼ 1.1� 106, three different points for the left (27, 38), right(77, 38), and central (50, 20) were selected to describe the periodic oscillations ofnumerical solutions (see Figure 7). If the solutions were axisymmetric, the valuesof temperature at the central point should have only one peak value, and the changeof the two-sided temperature should also be the same. However, this is not the case,as seen in Figure 7. In the duration of dimensionless time from 1,000 to 1,100, the

Figure 6. Periodic evolution of isotherms in a cycle at Ra¼ 106. (a) F¼ 2,001.62 or F¼ 2,017.44, (b)

F¼ 2,005.93, (c) F¼ 2,009.53, and (d) F¼ 2,013.84.

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first peak value of temperature at the central point is 0.150 at F¼ 1,007.06, while thesecond peak value is 0.155 at F¼ 1,014.86, as is shown in Figure 7. The mean valueof temperature at the right was about 0.416, and that of temperature at the left wasabout 0.367. The difference between them is about 13%. The numerical investigationrevealed that the natural convection turns to asymmetric periodical solution after ashort time, although the initial field and boundary conditions were symmetric. Morenumerical simulation indicated that the periodic oscillations of flow and heat trans-fer take place at a Rayleigh number varying from 4� 105 to 1.9� 106. In this scopeof periodic oscillation, an increase in Rayleigh numbers lead to the increase of themean value of Keq, as is shown in Figure 8. The maximum amplitude of Keq increasesgradually with Raleigh numbers, while the amplitude suddenly decreases at a criticalRayleigh number between 1.2� 106 and 1.3� 106.

Figure 7. Temperature time trace at different points for Ra¼ 1.1� 106.

Figure 8. Keq at different Rayleigh numbers. (a) Time-averaged value and (b) amplitude.

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4.3. On the Route to Chaos

The numerical solutions of velocity at the sample point (15, 30) and averagethermal conductivity Keq were obtained for several Rayleigh numbers from a lowvalue to 6� 106. The sample point is selected in order to analyze the nonlinear char-acteristics of the complicated system further. There is no special reason to select aparticular point since if the system is periodic it will be so at every point in the envel-oped cylinder. However, some points near boundary or in the middle may not besuitable due to small amplitude.

There are several ways to recognize the route to chaos: analyzing their time his-tory, phase space trajectories, and power spectral method. The effects of the increas-ing Rayleigh numbers on the route to chaos were analyzed by using phase spaceof velocity at the sample point and time history of Keq. A power spectrum ofHanning window provides an easy way to identify a chaotic system. The power spec-trum analysis of average thermal conductivity Keq is conducted to further study the

Figure 9. Time signal of Keq for (a) Ra¼ 4� 105, (b) Ra¼ 106, (c) Ra¼ 2� 106, and (d) Ra¼ 6� 106.

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route to chaos for the unsteady process. It can be seen from Figures 9a and 10athat the time signal is a constant for the average thermal conductivity, and the phasetrajectory of velocity at the sample point (15, 30) is a limit point for Ra¼ 4� 105. Asthe Rayleigh number increases further, a period doubling bifurcation occurred at aRayleigh number between 4.5� 105 and 6� 105. The time signal of Keq from zeroinitial fields is shown in Figure 9b. The data of dimensionless time less than 600, or1800 is not shown in this figure for the convenience of presentation. This limit pointloses its stability and allows the periodic solution to appear at Ra¼ 6� 105, as is seenin Figure 10b. The periodic solutions of velocity and thermal conductivity remainedfor the values of Rayleigh numbers from 6� 105 and 1.9� 106. In the periodic system,only one harmonic peak occurs, as shown in Figure 11b. A quasi-periodic solution ofthe system in Figures 9c and 10c was achieved for Ra¼ 2� 106. As the system pro-gresses towards chaos, more peaks will occur, associated with the harmonics,sub-harmonics and ultraharmonics of the system. There is one more bifurcation

Figure 10. Phase space of velocity at the point (15, 30). (a) Ra¼ 4� 105, (b) Ra¼ 106, (c) Ra¼ 2� 106,

and (d) Ra¼ 6� 106.

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and period doubling at successive critical values of Rayleigh numbers from 2� 106 to6� 106. For a truly chaotic system, there should be a continuous broadband spectrumin Figure 11d. It is ascertained in Figures 9d and 10d that periodicity is lost atRa¼ 6� 106. The results show that the oscillatory flow undergoes several bifurca-tions and ultimately evolves to a chaotic flow after the first bifurcation.

5. CONCLUSION

Numerical analyses of natural convection in a cylindrical envelope with aninternal concentric cylinder with slots have been presented. Governing equations

Figure 11. The power spectrum of Keq for (a) Ra¼ 4� 105, (b) Ra¼ 106, (c) Ra¼ 2� 106, and (d)

Ra¼ 6� 106.

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are solved using the SIMPLE algorithm with QUICK scheme. The internal slottedcylinder has been regarded as the isolated solid regions, and the integrative solutionmethod is used to solve the problem of isolated islands.

Flow and heat transfer characteristics are investigated for a Rayleigh numberrange of 4� 105 to 6� 106, while the Prandtl number is taken to be 0.71. When theRayleigh numbers are below 4� 105, the system is steady and strictly symmetricabout the vertical axis. For the cases with Rayleigh numbers greater than 6� 105,a doubling periodic time-dependent behavior is observed. For Ra¼ 1.1� 106, thenatural convection turns to a asymmetric periodical solution at after a short time,although the initial field and boundary conditions were symmetric. As the Rayleighnumber increases further, a quasi-periodic solution of the system was achieved atRa¼ 2� 106. There is one more bifurcation and period doubling at successive criticalvalues of Rayleigh numbers from 2� 106 to 6� 106. For higher Rayleigh numbersgreater than 6� 106, non-periodic oscillations were observed for the flow and heattransfer. The unsteady natural convection study presented here showed a transitionto the determinist chaos with increasing Rayleigh numbers.

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