Heat transfer in a "T" form cavity with heated rectangular blocks submitted to a vertical jet: The block gap effect on multiple solutions

Heat transfer in a ‘‘T’’ form cavity with heated rectangular blocks submitted to a vertical jet: the block

gap effect on multiple solutions

Mostafa Najam

, Mustapha El Alami, Abdelaziz Oubarra

Groupe Energeetique, Deepartement de Physique, Faculteedes Sciences, UniversiteeHassan II A€ıın Chock, Km 8 Route d’El Jadida, BP 5366, Ma^aarif Casablanca, Morocco

Received 3 June 2002; received in revised form 17 December 2002; accepted 24 May 2003

Abstract

A numerical investigation was conducted to study the enhancement of heat transfer in a cavity with heated rectangular blocks and submitted to a vertical jet of fresh air from below. The heated blocks are identical, and the system presents a symmetry with respect to a vertical axis passing through the middle of the opening. The cavity resembles a cooling passage of electronic equipment. The governing equations are solved by using the finite difference method. This study was conducted with the control parameters as:

Ra¼10⁵, 10⁶; 16Re61000; C¼0:15 and 0:256D60:75. The relative height of the blocks and the geometric parameter of the cavity are kept constant (B¼1=2 and A¼1). The results obtained, with Pr¼0:72, show the validity of multiple solutions on which the resulting heat transfer depends. Some useful correlations are then proposed.

2003 Elsevier Ltd. All rights reserved.

Keywords:Heat transfer; Vertical jet; Rectangular blocks; Multiple solutions

1. Introduction

Numerous studies related to mixed convection phenomena in channels have been reported in order to investigate the heat transfer and fluid flow in such geometries. This interest is dictated by its direct relation with the cooling of the components in the electronic industries. Because of the progress of circuit integration, the heat dissipation seems to be concentrated on fewer components

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*Corresponding author. Fax: +212-22-23-06-74.

E-mail addresses:najam@facsc-achok.ac.ma,mnejam@yahoo.fr (M. Najam).

0196-8904/$ - see front matter 2003 Elsevier Ltd. All rights reserved.

doi:10.1016/S0196-8904(03)00126-2

Nomenclature

A aspect ratio ðL

=H

Þ

B dimensionless block height ðh

=H

Þ C dimensionless opening diameter ðl

=H

Þ d

space between adjacent blocks

D dimensionless space between adjacent blocks ðd

=H

Þ h

block height

H

channel height l

opening diameter

L

length of calculation domain (cavity) n normal coordinate

Nu mean and normalized Nusselt number (Eq. (6)) Nu

local Nusselt number (Eq. (5))

Pr Prandtl number (Pr ¼ m=a) q

imposed flux

Ra Rayleigh number (Ra ¼ gbq

H

=ðamkÞ Re Reynolds number (Re ¼ v

H

=mÞ T

temperature of fluid

T dimensionless temperature of fluid ½¼ kðT

T

Þ=q

H

Þ u

; v

velocities in x

and y

directions

u; v dimensionless velocities in x and y directions ½¼ ðu

; v

Þ=v

v

average jet velocity at entrance (m/s)

x

; y

Cartesian coordinates

x; y dimensionless Cartesian coordinates ½¼ ðx

; y

Þ=H

a thermal diffusivity

b volumetric coefficient of thermal expansion k thermal conductivity of fluid

m kinematic viscosity of fluid q fluid density

w stream function

W dimensionless stream function ð¼ w=aÞ X

vorticity

X dimensionless vorticity ð¼ X

H

=aÞ Subscripts

0 dimensional variables c critical value, cold wall

f related to the forced convection H heated wall

max maximum

min minimum

ext extremum

as the system volume shrinks. The very advanced development of high density and large scale integrated chips have led to numerous miniaturized electronic devices. Consequently, heat from such sophisticated electronic components is excessively generated. A rigorous control of their operating temperatures appears indispensable, as it avoids their damage owing to an eventual overheat. In mixed convection, important works have been reported on problems related to the cooling of electronic components [1–3]. The recourse to this process of heat evacuation is justified by its simplicity and also by the low costs engendered. Generally, the components are disposed on horizontal cards and submitted to outside ventilation to evacuate the surplus of the generated heat. A literature review of recent works shows the interest and the realized progress in this field.

Furthermore, when the components are disposed on the internal plates of a horizontal channel, the space between the components remains not well ventilated, even in the case when the cooling process is ensured by a forced flow (FF) [4,5]. In fact, the FF, being parallel to the plane con- taining the components, often engenders recirculating movements between the blocks. Conse- quently, this region is not well ventilated.

Physical experiments and numerical simulations were performed by Leoni and Amon [6] on an embedded electronics prototype system of a wearable computer. Through the use of orthogonal arrays and optimal sampling, an efficient exploration of the parameter space was performed to determine thermal contact resistances and heat transfer coefficients. Bilen et al. [7] conducted an experimental and numerical investigation about the effect of the position of wall mounted rect- angular blocks on the heat transfer from the surface, taking into account the angular displacement of the block as well as its spanwise and streamwise disposition. The experiments were conducted in a rectangular channel with variable parameters as: distance between adjacent blocks, block displacement angle and Reynolds number. The results showed that the most efficient parameters were Reynolds number and angular disposition. The distance between blocks has a slightly increasing effect on the heat transfer. Murakami and Mikic [8] presented an optimization study using a method of determining optimum values of the channel diameter, flow rate and number of channels for minimum pressure drop. Various strategies have been explored to enhance the effectiveness of mixed convection cooling. These strategies include placement of an obstacle in the flow path of the coolant to destabilize the flow [9], using openings between blocks in recirculating movement spaces [10,11] or variable space length between two adjacent blocks [12].

Certainly, these strategies enhance the heat transfer between the blocks, but their vertical planes do not appear well ventilated when the jet is parallel to the block support. The aim of the present work is to send the jet perpendicular to the block support and to investigate the effect of space length between blocks on the mixed convective heat transfer. Thus, the configuration is a hori- zontal channel with rectangular heated blocks on its lower plane. The channel is submitted to a vertical jet of fresh air and rigid adiabatic planes are introduced to eliminate exchange of both heat transfer and matter between the different domains (cavities).

2. Physical problem and governing equations

A schematic representation of the configuration is depicted in Fig. 1a. Because of the problem

periodicity, the studied domain is reduced to a ‘‘T’’ form cavity, which is submitted to a laminar

vertical jet from below, Fig. 1b. The blocks are heated with a constant heat flux. The upper wall is cold, while the other sides of the cavity are insulated.

Using useful simplifications and current notations with the Boussinesq approximation, the non- dimensional governing equations in terms of vorticity X, stream function W and temperature T , are given by [13–15]:

oX ot þ ouX

ox þ ovX

oy ¼ Ra Re

Pr

oT ox

þ 1 Re

o

X ox

þ o

X oy

ð1Þ

oT ot þ ouT

ox þ ovT oy ¼ 1

RePr o

T

ox

þ o

T oy

ð2Þ

o

W ox

þ o

W

oy

¼ X ð3Þ

u ¼ oW

oy and v ¼ oW

ox ð4Þ

The dynamic and thermal boundary conditions are:

T ¼ 0; X ¼ 0; u ¼ 0; v ¼ 1 and W ¼ x 0:425 (at the admittance opening), T ¼ 0 (upper cold wall),

W ¼ 0 on the rigid walls situated to the left of the axis passing through the middle of the open- ings and W ¼ C on those to the right,

q ¼

¼ 1 (blocks with imposed flux).

At the evacuation opening, u; v; T ; X and W are extrapolated by adopting similar processes to those shown in Ref. [16,17] (the second spatial derivative terms in the vertical direction are equal to zero).

L’=H’ l’

Cavity H’

Micro cavity h’

Openings

d’

(a)

T=0

Forced flow (b)

x, u y, v

Constant flux

Adiabatic wall

g

l’

Fig. 1. (a) Model of two dimensional channel with openings; (b) calculation domain.

The local Nusselt number over the surface blocks (active walls) is calculated as:

Nu

¼ q

H

kðT

T

Þ ¼ 1

T ð5Þ

The mean dimensionless heat quantity produced by the blocks is:

Q ¼ Z

0

1 T

y¼0:5

dx þ Z

0

1 T

x¼a

dy þ Z

0

1 T

x¼b

dy þ Z

b

1 T

y¼0:5

ð6Þ with a ¼ 0:50 D=2 and b ¼ 0:50 þ D=2.

The mean normalised Nusselt number over the active walls is:

Nu ¼ Q Q

ð7Þ Q

is the mean heat quantity, calculated with Eq. (6) for Ra ¼ 0 and Re 6¼ 0.

3. Numerical method

For this problem, the governing equations were solved numerically using a finite difference technique. Central difference formulae were used for the discretization of all spatial derivative terms in the vorticity, energy and Poisson equations (respectively, Eqs. (1)–(3)). Diffusive terms are discretized using the upwind scheme. The final discretized forms of Eqs. (1) and (2) were solved by using the alternate direction implicit method. Values of the stream function were ob- tained with Eq. (3) via a successive over relaxation method [18]. At each time step, a variation by less than 10

over all the grid points for the stream function is considered as the convergence criterion. As a result of a grid independence study, a grid size of 41 · 41 was found to model accurately the flow fields described in the corresponding results. The time steps considered ranged between 10

and 10

.

The numerical model was validated with the benchmark solution of De Vahl Davis [19] relative

to the case of a differentially heated square cavity. Comparison of the obtained results shows

excellent agreement. The differences observed in terms of Nu and stream function values are less

than 2% for all the values of Ra considered. The program was also validated by comparing our

results against those of Y€ u ucel et al. [17] in the case of a channel of finite length discretely heated

from the bottom. The maximum relative deviations were found to be 1.6% for various combi-

nations of the governing parameters. Some supplementary tests were also done to verify the

energy balance of the system. Indeed, the total heat lost by the hot walls, should be equal to that

gained by the fluid inside the cavity. For all the considered cases, the overall energy balance

criterion was automatically checked by the numerical code. The integrated convective–conductive

heat flux through the exit reproduces the imposed conductive heat flux on the blocks with a

maximum relative difference of 0.2%. The accuracy of the numerical model was also verified by

comparing the results from the present study with those obtained by Amahmid et al. [20] for

natural convection in repetitive geometries. Good agreement was obtained in the W

term at low

Ra, but a slight difference appears at high Ra because of the use of different schemes (see Table 1).

This study is aiming to analyse the interaction between an external imposed flow, characterised by Re, and the natural convection developed in the cavity. We note that if the objective of this investigation were the chimney effect, the FF debit in the inlet would be a fundamental unknown of the problem, and we would have to determine it as a function of the control parameters (without Re).

4. Results and discussion

In this section, the heat transfer rates across the cold and the hot walls and the flow and temperature fields are examined for ranges of Reynolds number (1 6 Re 6 1000), distance between adjacent blocks ð0:25 6 D 6 0:75Þ, Rayleigh number (Ra ¼ 10

, 10

) and other parameters of the problem (B ¼ 0:5, C ¼ 0:15, A ¼ L=H ¼ 1, Pr ¼ 0:72).

The particularity of this problem is the solutions multiplicity when Re is less than a critical value (Re

). Essentially, the flow structure is composed of natural convection cells and forced convection open lines. The existence of three or four kinds of solutions is principally due to the conflict between the natural and forced convection. In the presence of one natural convection cell, the FF open lines pass by the left of the cell, Fig. 2a, or by the right of it (figure not presented).

The corresponding flows are, respectively, called, unicellular right or left flow (hereafter called UCRF or UCLF). These two solutions are reciprocal images in a vertical plane mirror (see the case of D ¼ 0:5). When two natural convection cells exist, the forced convection open lines pass between the cells or go a round of it. These two solutions are symmetrical, bicellular and, re- spectively, called intra-cellular and extra-cellular flows (called ICF and ECF), respectively, Fig. 2b and c. When the natural convection cells disappear, by increasing Re, we have only the FF in the cavity, Fig 2f.

4.1. Streamlines and isotherms

Two Rayleigh number values, sufficiently high, are selected in favour of the mixed convection role. Typical streamlines and isotherms will be presented for Ra ¼ 10

, 10

and various D and Re.

Hence, for Ra ¼ 10

and D ¼ 0:25, and at low Re (Re 6 10), we have obtained three solutions: the UCRF, Fig. 2a, its image UCLF (figure not presented) and the ICF, Fig. 2b. Only in this case ðD ¼ 0:25Þ, the UCLF is not steady. The UCRF is obtained by initialising the numerical code with dynamic field zero and linear thermal field in the vertical direction over the domain calcu- lation (called: conduction solution). Its image UCLF is obtained by inverting the W sign (it exists

Table 1

Model validation

Ra Najam (upwind) Amahmid (CDS)

10³ Wext¼0:024 Wext¼0:024

10⁴ W_ext¼0:77 W_ext¼0:80

10⁵ Wmin¼ 12:12 Wext¼9:49 Wmin¼ 13:64 Wext¼11 W_max¼8:93 W_max¼9:89

but is not steady in this case). We note that the corresponding isothermal distribution for the UCRF and UCLF shows that the evacuated heat flux is the same and is important in the left/

(right) part of the cold wall because of the contact of air heated by the block in the left/(right) of the cavity and the cold plane. The third solution is the ICF kind, Fig. 2b, symmetric with two natural convection cells above the blocks. The FF passes between the cells. This solution is ob- tained by initialising the code with a same solution, obtained for high Reynolds number (Re P 30). The closed cells attributed to natural convection are in direct contact with the cold wall. Consequently, the isotherm lines are very close near it. So, this solution seems to be the best of the lot for heat exchange. When Re exceeds 10, a fourth solution, ECF type, appears as pre- sented in Fig. 2c for Re ¼ 30. This solution is also symmetric, presenting two natural convection cells filling, approximately, all the available space above the blocks. The cells are contoured by the FF. For this solution, the numerical code is initialised by a dominating forced convection solution (Ra ¼ 0 and Re 6¼ 0). We note that the UCRF/(UCLF) and ICF solutions exist in the range 1 6 Re 6 65 (Re

¼ 65), and the ECF solution is absent at low Re values (Re 6 10) but existing in the range 10 < Re 6 65. When the intensity of the imposed flow increases (increase of Re), the convective cells become small and localised in the upper zone of the cavity. The space between adjacent blocks (called micro cavity) is filled with FF open lines as shown in Fig. 2c, d and e, respectively, for the ECF, UCRF and ICF solutions at Re ¼ 30. With further increase of Re up to Re

, the convective cells disappear, and the FF solution is installed, Fig. 2f, for Re ¼ Re

. The natural convection effect becomes negligible, and forced convection dominates.

For D ¼ 0:50 and Ra ¼ 10

, the major results are the disappearance of the ICF solution, the existence of the ECF one at low Re and the steady state of the UCLF solution. Thus, Fig. 3a–c present, respectively, the UCRF solution, its image UCLF and the ECF for Re ¼ 5. The cell sizes

(e) ICF

(d) UCRF

(b) ICF

(c) ECF (a) UCRF

(f) FF

Fig. 2. Streamlines and isotherms forRa¼10⁵;D¼0:25;Re¼10, (a) and (b);Re¼30 (c), (d) and (e);Rec¼65 (f).

increased because of the increase of the space between blocks. The vertical active planes (of the blocks) are more ventilated than before and the corresponding isotherms display good heat transfer. We note the existence of a rotating movement in the micro cavity (small cell) in the UCRF/(UCLF) case. In the case of the ECF solution, the natural convection cells are lying in the micro cavity. By increasing Re up to 10, there is, practically, no rotating movement, and the micro cavity is filled with the open lines for the UCRF/(UCLF) solution, Fig. 3d, Re ¼ 30. This solution disappears at Re

¼ 45.

For D ¼ 0:75 and Ra ¼ 10

, we have raised the same solutions as in the last case ðD ¼ 0:50Þ.

Fig. 4a and b present, respectively, the UCRF/(UCLF) and ECF solutions for Re ¼ 10. For Re ¼ 30, these solutions are given by Fig. 4c (UCRL/(UCLF)) and Fig. 4d (ECF). We note that the vertical active planes are, more and more, ventilated by increasing D and Re. We note that the UCRF/(UCLF) solution disappears at Re

¼ 35. The critical values of Re are displaced lower when D increases, and they are correlated with D as:

(b)

(c) (a)

(d)

UCRF ECF

Fig. 4. Streamlines and isotherms forRa¼10⁵;D¼0:75;Re¼10, (a) and (b);Re¼30, (c) and (d).

(c) ECF

(a) UCRF

(e) ECF

(d) UCRF

(b) UCLF

Fig. 3. Streamlines and isotherms forRa¼10⁵;D¼0:50;Re¼5, (a), (b) and (c);Re¼30, (d) and (e).

Re

¼ 30:25D

for Ra ¼ 10

In the case of Ra ¼ 10

, the flow structure is qualitatively similar to that obtained in the first case (Ra ¼ 10

): we have obtained the same kinds of solutions. Hence, for D ¼ 0:25, the UCRF/

(UCLF) solution exist (as earlier) in the range of Re between 1 and Re

(where Re

¼ 176), Fig. 5a, for Re ¼ 10. In the cases of the ICF and ECF solutions, there is a significant change of their existing domains. Fig. 5b (Re ¼ 10) presents a ICF kind of solution. It exists in the range of Re between 1 and Re

. This solution produce more heat than the UCRF/(UCLF) one. Unlike the first case (Ra ¼ 10

), the ECF exists at low Re (1 6 Re 6 25). As shown in Fig. 5c, for Re ¼ 10, the natural convection cells are not lying in the micro cavity. Consequently, this space is not venti- lated. The corresponding isotherms indicate that the heat transfer is weak in the horizontal di- rection from the vertical heated surfaces. We note that the ECF disappears at Re ¼ 25. We can note from Fig. 5d and e that there is a good heat transfer in the vertical direction from the horizontal heated planes and through the upper opening. We note also that the open lines con- strain the principal cells to deviate near the vertical adiabatic walls. In fact, in the range of Re between 10 and Re

, there is an important competition between the natural and forced convection for this value of Ra. When we approach Re

(Re

¼ 176), the cell sizes decrease, and at Re

, the FF occupied all the cavity, as in the case of Ra ¼ 10

(figure not presented). The shear effect on the jet due to the micro cavity corners, intensified by the presence of the cells, is probably the reason for the instability of the ECF solution. Certainly, this shear effect becomes negligible by increasing D.

Hence, for D ¼ 0:5, the ICF solution does not exist, and contrarily, the ECF existence domain is larger than earlier. On the other hand, for Re ¼ 10, the space offered by increasing D permits a rotating movement in the micro cavity, in the case of the UCRF/(UCLF), Fig. 6a. The size of the natural convection cell increased and is filling all the available space above the blocks. The cor- responding isotherms show that there is an amelioration of heat transfer from the vertical heated surfaces. For the ECF solution, Fig. 6b, the natural convection cells are lying in all the cavity. The

(d) UCRF (c)

ECF

(e) ICF

(a) UCRF

(b) ICF

Fig. 5. Streamlines and isotherms forRa¼10⁶;D¼0:25;Re¼10, (a), (b) and (c);Re¼30 (d) and (e).

increase of Re reduced the cell sizes as shown in Fig. 6c (UCRF/(UCLF)) and Fig. 6d (ECF) for Re ¼ 30. The jet becomes relatively intense and opposite to the fresh air descent toward themicro cavity in the case of the ECF solution. Consequently, the heat transfer becomes poor in this zone as shown by the temperature field. At Re ¼ Re

(Re

¼ 137), only the ECF solution exists with small convective cells, which disappear just after Re

, and the FF appears. For D ¼ 0:75, we have obtained the same solutions as in the case of D ¼ 0:5, Fig. 7a and b. The rotating movement in the micro cavity (case of UCRF/(UCLF)) becomes visible because of the widening of this zone. The heat transfer from the vertical active planes is ameliorated. With further increase of Re, up to Re

(Re

¼ 95), the natural convection cells resist the jet only in the case of the ECF solution (fig. not presented). Finally, we note that the critical value of Re (Re

) increases with Ra and decreases with D. It is correlated by

Re

¼ 86:10D

for Ra ¼ 10

(c) (d)

(f) (e)

(a) (b)

UCRF ECF

Fig. 7. Streamlines and isotherms forRa¼10⁶;D¼0:75;Re¼10, (a) and (b);Re¼30 (c) and (d);Re¼70, (e) and (f).

(d) ECF (a)

UCRF (b)

ECF

(c) UCRF

Fig. 6. Streamlines and isotherms forRa¼10⁶;D¼0:50;Re¼10, (a) and (b);Re¼30 (c) and (d).

4.2. Heat transfer

The mean dimensionless heat quantity leaving the active planes of the blocks is calculated by Eq. (6). Its variation with Re is presented by the mean and normalised Nusselt number (Eq. (7)) for different values of D as shown in Fig. 8 for Ra ¼ 10

and Fig. 9 for Ra ¼ 10

. Generally, these figures show that the Nu presents two or three curves in the range of Re between 1 and Re

(zone of multiple solutions). We note that the Nusselt number is practically constant at low Re (1 6 Re 6 10). In this range, the heat transfer is essentially by natural convection. We note that the Nu increases with D. This is probably due to good ventilation of the vertical active walls, which

1 10 100 1000

0.8 1.2 1.6 2

Nu

Re

UCRF/(UCLF) ECF ICF

D=0.75

D=0.50

D=0.25

Fig. 8. Nusselt variation withReforRa¼10⁵ and different values ofD.

1 1 0 100 1000 0.00

1.00 2.00 3.00 4.00

N u

Re U C R F / ( U C L F ) ECF ICF

D=0.75

D=0.50

D=0.25

Fig. 9. Nusselt variation withReforRa¼10⁶ different values ofD.

leads to the blocksÕ temperature reduction. We note that these vertical planes form 75% of the active surfaces.

For Ra ¼ 10

and D ¼ 0:25, Fig. 8, we have three curves in the range of Re1 6 Re 6 Re

(Re

¼ 65). The upper branch is related to the ICF solution. The middle branch is engendered by the UCRF/(UCLF) solution, and the lower branch, existing in a limited range of Re (10 6 Re 6 Re

), is related to the ECF solution. For Re between 10 and Re

, the Nu presents a rapid decrease because of an important competition between natural and forced convection. Up to Re

, the forced convection dominates, and Nu decreases to 1 for Re ¼ 200. For D ¼ 0:5, Nu presents the same profile as the first case ðD ¼ 0:25Þ, but we have only two curves in the multiple solutions zone (1 6 Re 6 Re

, Re

¼ 45) because of the absence of the ICF solution. We note that Nu is constant in the range of Re between Re

and 100, and then decreases to 1. For D ¼ 0:75, the mean Nusselt number is more important than before, and the only particularity is that Nu increases when Re is in the range Re

6 Re 6 100 and then decreases rapidly to 1.

For Ra ¼ 10

, Fig. 9 shows that the Nu profile is similar to that of the first case (Ra ¼ 10

). For D ¼ 0:25, we have always three curves, but their existing domains are changed. Hence, the branch related to the ECF solution (lower branch) exists in the range of Re 1 6 Re 6 25. The UCRF/

(UCLF) branch is, as earlier, in the range 1 6 Re 6 Re

(Re

¼ 176), and the upper branch, related to the ICF solution, exists in the domain of multiple solutions (1 6 Re 6 Re

). In the other case, D ¼ 0:5, the heat transfer presents a different behavior. So, for Re between 10 and 40, the ECF solution produce heat, only in this case more than the UCRF/(UCLF) solution. For Re > 10, Nu decreases rapidly along the upper branch (ECF), and so, there is an overlapping at Re ¼ 40, after which the ECF branch becomes lower in the range of Re 40 6 Re 6 Re

(Re

¼ 95). Up to Re

, Nu decreases regularly to 1 (forced convection) for Re ¼ 1000. Finally, for D ¼ 0:75, the Nu variation with Re is the same as in the case of Ra ¼ 10

.

5. Conclusion

A numerical study of laminar mixed convection in a ‘‘T’’ form cavity submitted to a vertical jet was conducted. For a given Re, the study shows that there exist multiple solutions of the problem on which the resulting heat transfer depends significantly. These solutions disappear when Re achieves a critical value, which is correlated with D as Re

¼ 30:25 D

, for Ra ¼ 10

and Re

¼ 86:1 D

for Ra ¼ 10

. The mean normalised Nusselt number (Nu) increases with D and decreases with Re. ItÕs a maximum and practically constant if Re is less than 10. In this range of Re, the heat transfer is essentially by natural convection. When the ICF solution exists, it performs a good ventilation of all the active planes of the cavity, and the ECF solution is not useful for heat evacuation, since it leads to the lower of the mean number of Nusselt. Finally, it is noteworthy that the heat transfer is enhanced by increasing the space between the blocks.

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