SIMULATIONS OF THE KETTLE REBOILER SHELL SIDE THERMAL-HYDRAULICS WITH DIFFERENT … · ·...

SIMULATIONS OF THE KETTLE REBOILER SHELL SIDE

THERMAL-HYDRAULICS WITH DIFFERENT TWO-PHASE

FLOW MODELS

by

Milada PEZO, Vladimir D. STEVANOVI], and @arko STEVANOVI]

Original scientific paperUDC: 532.529:66.011

BIBLID: 0354-9836, 10 (2006), 2, 127-140

A com pu ta tional fluid dy nam ics ap proach is pre sented for the sim u la tionand anal y ses of the ket tle reboiler shell side ther mal-hy drau lics with twodif fer ent mod els of two-phase flow – the mix ture and two fluid model. Themix ture model is based on solv ing one mo men tum equa tion for two-phasemix ture flow and a clo sure law for the cal cu la tion of the slip be tween gasand liq uid phase ve loc i ties. In the two fluid mod el ing ap proach the mo men -tum bal ance is formed for each phase, while the gas-liq uid in ter ac tion dueto mo men tum ex change at the in ter face sur face is pre dicted with an em pir i -cal cor re la tion for the in ter face fric tion co ef fi cient. In both ap proaches thetwo-phase flow is ob served as two inter-pen e trat ing con tinua. The mod elsare solved for the two-di men sional ge om e try of the ket tle reboiler shell sidever ti cal cross sec tion. The com pu ta tional fluid dy nam ics nu mer i cal methodbased on the SIMPLE type al go rithm is ap plied. The re sults of both liq uidand vapour ve loc ity fields and void frac tion are pre sented for each mod el -ing ap proach. The cal cu lated void frac tion dis tri bu tions are com pared withavail able ex per i men tal data. The dif fer ences in the mod el ing ap proachesand ob tained re sults are dis cussed. The main find ing is that the void frac -tion dis tri bu tion and two-phase flow field strongly de pends on the mod el ingof the slip be tween liq uid and gas phase ve loc ity in mix ture model or on thein ter face fric tion model in two fluid model. The better agree ment of the nu -mer i cally pre dicted void frac tion with the ex per i men tal data is ob tainedwith the two fluid model and an in ter fa cial fric tion model de vel oped for thecon di tions of two-phase flows in large vol umes of ket tle reboilers or dif fer -ent de signs of steam gen er a tors.

Key words: kettle reboiler, thermal-hydraulics, numerical simulation,computational fluid dynamics

Introduction

Shell and tube heat exchangers are among the most widely used types of heatexchangers. Var i ous shell and tube heat exchangers are de signed for vapour gen er a tionon the shell side. They are widely ap plied in chem i cal, pro cess, and en ergy power in dus -try, in refregirations and air-con di tion ing equipments, and they are ap plied such asreboilers, steam gen er a tors, and evap o ra tors. It has been es ti mated that more than 50% of

127

all heat exchangers em ployed in pro cess in dus tries are used to boil flu ids and in volvetwo-phase flow on the shell side [1]. In pro cess in dus try they are known as reboilers,while ket tle reboilers are one of the most com mon reboiler types [2]. Also, some de vel op -ments of hor i zon tal steam gen er a tors for nu clear power plants are based on the ket tlereboiler de sign [3].

A typ i cal de sign of the ket tle reboiler ap plied in the pro cess in dus try is shown infig. 1. The evap o rat ing fluid flows on the shell side, across a hor i zon tal tube bun dle. Theheat is trans ferred to the boil ing two-phase mix ture from a hot fluid that cir cu lates in sidethe tubes. The liq uid level is con trolled by a weir, so that the bun dle is al ways sub mergedin liq uid. The gap be tween the bun dle and the shell al lows in ter nal recirculation of liq uid. The liq uid en ters the bun dle at its bot tom only. The mass ve loc ity of fluid across the bun -dle is in creased by the recirculation of liq uid, af fect ing the global heat trans fer co ef fi -cient.

Pre vi ous in ves ti ga tions of the ket tle reboiler shell side ther mal-hy drau lics havebeen per formed with ex per i men tal and an a lyt i cal mod els of var i ous lev els of com plex ityre gard ing the multidimensionality and ther mal-hy drau lic com plex ity of boil ing two--phase flow con di tions. Void frac tions and pres sure drops in one-di men sional up wardtwo-phase flows across the tube bun dles with var i ous tubes’ ar range ments are pre sentedin [4, 5]. One di men sional in ves ti ga tions were per formed first, be cause the pro cess ofevap o ra tion is com plex and the ver ti cal flow across the tube bun dle is dom i nant. But,reboiler shell side ther mal-hy drau lics is strongly in flu ence by mul ti di men sional ef fects.It was shown that the ver ti cal pres sure change is not con stant along the bun dle, caus ing alat eral pres sure change, which must be sat is fied by a lat eral flow [6]. In [7] a more re al is -tic two-di men sional in ves ti ga tion is pre sented with in for ma tion how vari a tion of heatflux, weir height, bun dle size and pres sure af fect the ket tle reboiler pro cesses. Two-di -men sional nu mer i cal mod els of the ket tle reboiler shell side ther mal-hy drau lics are pre -sented in [8-10]. The main find ings in per formed re searches are: (a) the ho mo ge neousmodel of two-phase flow pro vides too high val ues of the void frac tion which means thatthe gas and liq uid phase ve loc ity slip should be taken into ac count in a ket tle reboiler de -

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THERMAL SCIENCE: Vol. 10 (2006), No. 2, pp. 127-140

Figure 1. Schematic viewof the kettle reboiler

sign or anal y ses pro ce dures, and (b) the liq uid phase cir cu la tion is or ga nized on thereboiler shell side, where the intensity of circulation influences the heat transfercoefficient and the void fraction distribution.

This pa per pres ents the pos si bil i ties of two com monly ap plied two-phase flowmod els for the pre dic tion of the ket tle reboiler shell side ther mal-hy drau lics. These are(a) the mix ture model of two-phase flow with the ap pli ca tion of the clo sure law for thepre dic tion of the gas and liq uid phase slip ve loc ity (for in stance pre sented in [11]), and(b) the two fluid model of two-phase flow with the clo sure law for the pre dic tion of thegas and liq uid phase in ter fa cial friction (applied in [10]).

Modelling approaches

Sev eral as sump tions are in tro duced in mod el ing the ket tle reboiler shell sidether mal-hy drau lics:

(a) The shell side flow in the slab of the kettle reboiler vertical cross section (as presentedwith the left scheme in fig. 1) is two-dimensional;

(b) Steady-state conditions are modelled;(c) The shell side two-phase mixture is saturated;(d) The surface tension at the gas-liquid interface is neglected, as it is not important for

bulk two-phase flow phenomena. Hence, pressure is the same for both phases withinthe numerical control volume;

(e) Flow governing equations are written in the non-viscous form, while the turbulentviscosity effects are taken into account indirectly through friction coefficients for thetube bundles flow resistance and two-phase interfacial friction force; and

(f) The porous medium concept is used in the simulation of two-phase flow within tubebundles. In the applied mixture model it is assumed that the porous media is 100%open to the fluid flow, while in the two-fluid model the real volumes of liquid and gasphase are taken into account. In both models, the resistance of the tube bundle to thetwo-phase flow is taken into account through the appropriate volumetric force.

Mixture model

The mix ture model solves the con ti nu ity equa tion for the mix ture, the mo men -tum equa tion for the mix ture, and the vol ume frac tion equa tion for the sec ond ary phases,as well as al ge braic ex pres sions for the rel a tive ve loc i ties (if the phases are mov ing at dif -fer ent ve loc i ties) [11]. The en ergy equa tion for the mix ture is also a con stit u ent of themix ture model, but here it is not in cluded since the sat u rated liq uid and two-phase flowson the reboiler shell side are considered.

The con ti nu ity equa tion for the two-phase mix ture is:

Ñ × =( )rm vr

m 0 (1)

129

Pezo, M., Stevanovi}, V. D., Stevanovi}, Z.: Simulations of the Kettle Reboiler Shell ...

where rvm is the mass-averaged velocity:

rr

vv

f g

mm

kk k k

=å=

a r

r(2)

and rm is the mixture density:

r a rmf g

= å=k

k k (3)

ak is the volume fraction of phase k (liquid phase k = f or gas phase k = g).The mo men tum equa tion for the mix ture can be ob tained by sum ming the in di -

vid ual mo men tum equa tions for both phases:

Ñ × = -Ñ + + Ñ × -( ) ( )r r a rm m g g g gv v g v v Fr r r r r r

m m dr dr wmp (4)

where rv gdr is the drift velocity of the dispersed vapour phase:

r r rv vg gdr mv= - (5)

In tro duc ing eq. (2) in eq. (5) the fol low ing ex pres sion for the vapour drift ve loc -ity is de rived in which the slip ve loc ity be tween the va por and liq uid phase fig ures (alsore ferred to as the rel a tive ve loc ity):

r r rv v vf g g f= - (6)

The slip ve loc ity is cal cu lated with the fol low ing semi-em pir i cal cor re la tion:

r rv

18af g

m g g2

f drag

=-( )r r

m

d

f(7)

where dg is the bubble diameter and ra is the mixture acceleration. The drag coefficient

fdrag is calculated with the empirical relations of Schiller and Naumann:

f drag =+ £

>

ìíî

1 015 1000

00183 1000

0 678. Re , Re

. Re, Re

.

(8)

and the acceleration ra is of the form:

r r r rr

rr

a g v vv

gDv

D= - ×Ñ - = -( )m m

m m

t t

¶

¶(9)

The vol u met ric force of tube bun dle re sis tance to two-phase mix ture flow rFwm

is cal cu lated as:

r rF

em mwm

ee

ev

e= å

=1

2 2

2V

r

D(10)

130


where re is a unit vector in the direction of the Cartesian coordinate axis, and De is the

width of the computational cell. The coefficient of the local pressure drop in e direction Ve

is calculated according to [12]. The vapour vol ume frac tion is cal cu lated from the con ti nu ity equa tion:

Ñ × = -Ñ ×( ) ( )a r a rg g g g gv vr r

m dr (11)

Two fluid model

Here ap plied two-fluid model of two-phase flow con sists of the con ti nu ity andmo men tum equa tions for each phase and cor re spond ing clo sure laws.

Mass con ser va tion equa tions for liq uid and vapour phase are:

¶

¶

( )( )

a ra rf f

f f fUt

e+ Ñ × = -r

G (12)

¶

¶

( )( )

a ra r

g gg g gU

te+ Ñ × =

rG (13)

Liq uid and vapour mo men tum con ser va tion equa tions:

¶

¶

( )( )

a ra r a a rf f f

f f f f f f f g f fU

U U g + F F

rr r r r r

tp w+ Ñ × = - Ñ + - - Ge i

rU f (14)

¶

¶

( )( )

a ra r a a r

g g gg g g g g g g g f g

UU U g + F F

rr r r r r

tp w+ Ñ × = - Ñ + - - Ge i

rU f (15)

Evap o ra tion rate is cal cu lated as:

Ge

q

ha= cell

fgw (16)

where q A qcell cell cell= ¢¢ (17)

when heat flux at the wall of the shell tube is defined, or:

q h A Tcell tp cell sat= D (18)

when the temperature of the tube wall is defined. The heat transfer co ef fi cient cal cu la tions in the tube bun dle are de fined by the

Chen type of cor re la tion [13]:

h Fh Shtp conv nb= + (19)

131


or

hk

D F

k cp

tp

f f f

= +

+

0023

000122

0 8 0 4

0 79 0 45 0 49

.

Re Pr

.

. .

. . .r

s m r0 5 0 29 0 24 0 240 24 0 75

. . . .. .( ) ( )

f fg gw sat w

hT T p p S- -

(20)

where applied coefficients are defined as:

p pw sat= (21)

F X

F X XTT

TT

= £

= +

-

-

10 010

235 0213

1

1 0 736

. .

. ( . ) .

for

for TT-

>1

010.(22)

S F F

S

= + <

= +

-[ . (Re ) ] Re .

[ . (

. . .1 012 325

1 0 42

1 25 114 1 1 25for

Re ) ] . Re .

. Re

. . .

.

F F

S F

1 25 0 78 1 25

1 25

325 700

01 7

for

for

£ <

= ³ 00.

(23)

and Martinelli parameter:

X TT =-æ

èç

ö

ø÷

æ

èçç

ö

ø÷÷

æ

è

çç

ö

ø

÷÷

10 9 0 5 01

a

a

r

r

m

m

. . .

g

f

f

g

(24)

and Reynolds number:

Re( )

=-G X Dhy1

m f

(25)

The wall drag term can be ex pressed as a pres sure drop due to the tube bun dlefric tion. The pres sure drop due to the two-phase mix ture flow around tubes in a bun dle isde ter mined by tak ing into ac count the sep a rate con tri bu tion of the each phase to the to talpres sure drop. The pres sure drop of phase k in e di rec tion is defined as:

r rF

ew w

f gk k e

k k e ku

e= -

+V

ra

a

a a

$( )

2

21

D(26)

where Vk e is pressure loss coefficient in y direction and $ , ,u kk e = f g is the maximumvelocity of the phase in e direction.

The in ter fa cial fric tion terms have in flu ence on void frac tion and rel a tive ve loc -ity of the phases [14]:

r r r r rF U U U Ug f f

bg f g f= × - -

3

4r a

C

D

d i( ) (27)

132


where Cdi is the interfacial drag coefficient and Db is the diameter of the bubble. Forbubble flow j £ 0.3 [14]:

C Dd i =+ì

íï

îï

üýï

þï0267

1 1767

1867

672

.. ( )

. ( )b

g f

f

Dr

s

j

j(28)

wheref ( ) ( ) .j j= -1 1 5 (29)

and

ja

a a=

+

g

f g

(30)

For churn-tur bu lent flows (j > 0.3), a new cor re la tion is pro posed based on there sults pro posed in [15]:

C Dd i = - -1487 1 1 0 753 2. ( ) ( . )bgDr

sj j (31)

The func tional de pend ence of the ra tio Cdi/Db on the void frac tion is given in fig. 2. For tran si tional flow pat terns, there is a rapid de crease of Cdi/Db ra tio. This could be at -trib uted to the de crease of the two-phase flow in ter fa cial area con cen tra tion.

According to the ap plied po rous me dia ap proach, the con trol vol ume within thetube bun dle is oc cu pied with tubes and a free area filled with sin gle phase or two-phasemix ture. Tube bun dle is rep re sented as po rous area, with the porous fac tor:

y = -1 4

2

2

D

a

p(32)

The pre sented mod els are solved with the con trol vol ume based nu mer i calmethod and code pre sented in [16]. Details about the solv ing pro ce dure are pre sented in[17].

133


Figure 2. Ratio of interfacial frictioncoefficient Cdi and particle diameter Dp vs.void fraction for two-phase flow across a tube bundle

Boundary conditions

Bound ary con di tions in clude the line of sym me try, the shell wall, and the in let flowsand out let re cir cu lat ing flow. At the line of sym me try gra di ents of all vari ables are con -stant. Sat u rated liq uid in flow was spec i fied at the bot tom cen ter of the heat exchanger,sim u lat ing the feed flow in an ac tual ket tle reboiler. The flow rate was spec i fied us ing theover flow and the va por iza tion rate. At the out flow bound ary in an ac tual ket tle reboiler,the vapour sep a rates from the liq uid and leaves the reboiler. In the pro cess, most of the

liq uid recirculates within the reboiler, but some of the liq uid recirculates within the evap o ra tor, andsome of the liq uid is car ried over the weir.Recirculation of the liq uid is sim u lated by de fin -ing the ve loc ity gra di ent at the out let sur face.Bound ary con di tions are pre sented in fig. 3 fortwo fluid model ve loc ity com po nents V along ver -ti cal co or di nate and W com po nent along hor i zon -tal co or di nate. The re duc tion of these bound arycon di tions to the mix ture model is straight for -ward. In the mix ture model the void frac tion at the free sur face of the two-phase mix ture is equal to 1, while no change of the two-phase mix ture ve loc -ity com po nents is as sumed as in case of thevapour ve loc ity com po nent in the two fluid model (fig. 3).

Results and discussion

De vel oped mod els of two-phase flow on the ket -tle reboiler shell side are ap plied to the nu mer i calsim u la tion and anal y ses of the ex per i men tal con di -tions pre sented in [18]. Geo met ric pa ram e ters re -quired to spec ify the heat exchanger are shown in fig. 4 and tab. 1. Ex per i ments were per formed un derat mo spheric pres sure.

The pre sented reboiler ge om e try is discretizedwith two-di men sional con trol vol umes, fig. 5.

Re sults of the nu mer i cal sim u la tions per formedwith two fluid model and mix ture model are pre -sented in figs. 6, 7, and 8. In those pic tures to taltwo-phase flow mass flux vec tors were cal cu latedas:

r r rG f f f= +a r a rU Ug g g (34)

134


Figure 3. Boundary conditions

Figure 4. Geometry of the kettlereboiler experimental test section[18]

The re sults in fig. 6 clearly show thefor ma tion of one cir cu la tion cen tre at thebound ary of the up per half of the tubebun dle (the bun dle is de picted withdashed line). Strong down ward flow ex -ists in the downcomer be tween the bun dle and the shell wall. This nat u ral cir cu la -tion is gov erned by the den sity dif-ferences be tween the mix ture in the bun -dle-shell downcomer (lower void frac tion and higher den sity) and the boil ingtwo-phase mix ture in the bun dle (highervoid and lower den sity). The ref er encemass flux vec tors of 1400 kg/m2s for twofluid model re sults and 1700 kg/m2s incase of mix ture model in di cate al most thesame in ten sity of the over all cir cu la tionaround and through the bun dle. But, themix ture model pre dicts less in ten sive lat -eral flow from the downcomer to wards the tube bun dle. Ac cord ing to the mix ture model,the two-di men sional char ac ter of the flow be tween the downcomer and the tube bun dle is sup pressed; both the bun dle and downcomer flows are ver ti cal in op po site di rec tions.

The two-phase flow model pre dicts lower void frac tions than the mix ture model, fig. 7. This is due to the more in ten sive liq uid down ward flow in the bun dle-shelldowncomer and liq uid pen e tra tion from the downcomer to the bun dle, as pre sented in thecor re spond ing left pic ture in fig. 7. The thick full line in both pic tures in fig. 7 rep re sents

135


Table 1. Dimensions and parameters of the experimental kettle reboiler [18]

Tube diameter 2r 19 mm

Pitch h 25.4 mm

Shell radius R 0.368 m

Weir height from thecenter of shell b

0.210 m

Bundle center/shellradius offset a

0.114 m

Number of tubes 241

Tubes arrangement In-line square

Working fluid Re frig er ant R113Figure 5. Meshes used for modeling

Figure 6. Mass flux vector for liquid for twofluid model (left) and mixture model (right)

the lower bound ary of the re gion of highervoid as ob served in the ex per i men tal in ves ti -ga tion. The re gion above the full thick line is char ac ter ized as frothy. In right side of fig. 7a very rapid in crease in void is cal cu latedabove the di vid ing line with void val uesabove 0.8 at the top of the bun dle. It is hardly to ex pect that the high val ues of void frac -tion can be achieved un der con sid ered heatflux of 20 kW/ m2.

Fig ure 8 shows the hor i zon tal void frac -tion dis tri bu tions at the half of tube bun dleheight cal cu lated with the Scharge et al. em -pir i cal cor re la tion as re ported in [9] and pre -dicted with here pre sented two fluid modeland mix ture model. The mix ture model doesnot pro vide sat is fac tory agree ment with thetwo fluid model and the ref er ent data ofSchrage, es pe cially in the in ner ar eas of thebun dle close to the axis of sym me try. The

two fluid model pro vides much better agree ment with the ref er ent void frac tion data.As it is pre sented with the re sults of the nu mer i cal sim u la tion, the mix ture model

is less suit able for the sim u la tion and anal y ses of the ket tle reboiler shell side ther mal-hy -drau lics. This is at trib uted to the fact that the mix ture model is in her ently not suit able forthe mod el ling of vapour and liq uid phase sep a ra tion in the large two-phase vol umes withliq uid phase recirculation, es pe cially at the two-phase mix ture free-sur face (so calledswell level).

136


Figure 7. Void fraction contour plot fortwo-phase model (left) and mixture model(right) with comparation with Cornwellsexperiments [15]

Fig ure 8. Com par i son of thehor i zon tal void frac tion dis tri -bu tions at the half of the tubebun dle height cal cu lated withthe Scharge et al. model and re -ported in [9] and pre dicted withhere pre sented two fluid modeland mix ture model

Conclusions

The mix ture and two fluid mod els are ap plied to the sim u la tion of the ket tlereboiler shell side ther mal-hy drau lics. Re sults ob tained with the mix ture, based on thetwo-phase mix ture con ti nu ity and mo men tum equa tion and cor re spond ing semi-em pir i -cal clo sure laws for the va por and liq uid phase slip ve loc ity, does not pro vide re li able re -sults of the ket tle reboiler shell side ther mal-hy drau lics. The lat eral two-phase flow fromthe downcomer to the tube bun dle is sup pressed and too high val ues of the void frac tionare ob tained. On the other hand, more com plex two fluid model, based on the mass andmo men tum bal ance equa tions for each phase and cor re spond ing clo sure laws for the in -ter face mo men tum ex change due to the in ter face fric tion, pro vides much better agree -ment with the ref er ent re sults of the void frac tion dis tri bu tion. Also, two fluid model pro -vides more plau si ble re sults of the two-phase flow struc ture on the ket tle reboiler shellside than the mix ture model. The in abil ity of the mix ture model to pre dict cor rectly theshell side ther mal-hy drau lics is due to the solv ing of one mo men tum equa tion for thetwo-phase mix ture, and cor re spond ing lim i ta tions in mod el ling the vapour and liq uidphase sep a ra tion within the large volume of the kettle reboiler shell side with liquid phase cir cu la tion.

Nomenclature

A – area of tube bundle cell, [m]ra – mixture acceleration, [ms–2]aW – specific area of tube bundle cell, [m–1]Cdi – interfacial drag coefficient, [-]cp – specific heat at constant pressure, [kJkg–1K–1]D – tube diameter, [m]Db – diameter bubble, [m]Dhy – hydraulic equivalent diameter, [m]dg – bubble diameter, [m]re – unit vector in the direction of the Cartesian coordinate axis, [–]De – width of the computational cell in y or z direction, [m]rF – source term, [Nm–1]Fwm – volumetric force of tube bundle resistance to two-phase mixture, [kgm–2s–2]fdrag – drag coefficient, [-]G – mass flux, [kgm–2s–1]g – gravitational acceleration, [ms–2]h – heat transfer coefficient, [Wm–2K–1]k – thermal conductivity, [kWm–1K–1]Pr – Prandtl number, (= mcp/k) [-]p – pressure, [Pa]q – heat flux, [Wm–2]Re – Reynolds number, (= rUDhy/m), [-]T – temperature, [K]t – time, [s]U – velocity of the phase (components Uky = Vk, Ukz = Wk), [ms–1]

137


v – velocity, [ms–1]rv dr – drift velocity, [ms–1]rv m – mass-averaged velocity, [ms–1]rv fg – relative velocity, [ms–1]X – flow quality, [–]XTT – Martinelli parameter, [–]

Greec letters

a – volume fraction in the control volume, [–]Ge – evaporation rate, [–]V – pressure loss coefficient, [Nm–1]s – surface tension, [Nm–1]m – viscosity, [Pas]r – density, [kgm–3]j – vapour volume fraction (void) in two-phase flow, [–]y – porosity, [–]

Subscripts

conv – convectione –

re direction

f – liquidg – gasi – interface parameterin – inletk – phasem – mixturenb – neigbour cellp – particlesat – saturationtp – tube bundlew - wally - pertain to y directionz - pertain to z direction

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138


[6] King, M. P., Jensen, M. K., Lo cal Heat Transfer and Flow Pattern Distributions in a KettleReboiler, Two-Phase Flow Mod el ling and Ex per i men ta tion, (1995), pp. 1289-1296

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Authors' addresses:

M. Pezo, @. Stevanovi}VIN^A Institute of Nuclear Sciences,Department of Thermal Engineering and EnergyP. O. Box 522, 11001 BelgradeSerbia

V. D. Stevanovi}Faculty of Mechanical Engineering, University of Belgrade16, Kraljice Marije, 11000 BelgradeSerbia

Corresponding author (V. D. Stevanovi}):E-mail: [email protected]

Paper submitted: January 18, 2006Paper revised: June 10, 2006Paper accepted: June 15, 2006

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