Different Solver methods used in Zonal Models

8/18/2019 Different Solver methods used in Zonal Models

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Solver used in

Zonal models



Introduction

ZONAL MODEL• The term. “Zonal model", was rst !ro!osed # Lebrun in his

in %&'( )Allard and Inard, %&&*+.

• Zonal model is a sim!lied model and it is the intermediate o t

room model and /D model.

• It satises the necessar# to evaluate the local thermal comort a

air 0ualit# 1ithin various 2ones o the uildin3.

• Zonal model !rovides the detailed 4no1led3e o air5o1 and tem

distriution 1hich 1as ailed to !rovide # the Multi-room modeeas# to use and 1ill not su6er rom hi3h com!utation li4e /D m



Limitations o !revious mode

7$revious 4no1led3e o air 5o1 !attern is re0uired.

7 The# cannot !redict the tem!erature variation at the hori2olevel.

7 The# have een develo!ed or s!ecic a!!lications onl#.



8asic Assum!tions

7 The air is inviscid.

7 The air tem!erature and densit# are uniorm.

7 The air !ressure in the 2one is assumed to e h#drostaticaldistriuted ased on the reerence !ressure.

7$rinci!le o conservation o Mass and Ener3# must e oe#

each 2one.79adiation com!onent is i3nored.

7 Three 4inds o oundaries are identied.7 Normal oundar#

7 :et oundar#

7 Mi;ed oundar#



Techni0ues used in !revious2onal models7 Three cate3ories o irre3ular cells, that can e distin3uished 1ithin a room.

7 8ouia )%&&<+ = Lo1 velocit# cells.

7 Allard et al. )%&&( and %&&>+ = Drivin3 cells.

7 Inard et al. )%&&?+ = Mi;ed cells.

7 /or lar3e indoor s!aces, 8ouia and @urt2)%&&+ initiated develo!ment methods solvin3 the !ressure eld to !redict air5o1 and tem!erature.

7 @urt2= Bses mass and ener3# alances are 1ritten in each suvolume, 1hile the the interaces are calculated # !o1er-!ressure la1s.

7 Coelt2el)%&&&+= To !redict the air 5o1 !attern and tem!erature elds in Atria, shela1 5o1 e0uation 1ith constant coecients.



onti..

7 Muss#)%&&&+= She entirel# reormulates the 2onal model develo!ed # @as a connection o small sets o e0uations in a 1a# that all the e0uationsthe uildin3 ehavior are 3rou!ed to3ether into sus#stems o e0uation

7 a3hi3hat et al. )*((%+= S!ecic di6erences occur in the treatment o mrates usin3 a !ressure !o1er la1 versus tem!erature and heat !o1er la

7 The# !rovided a com!rehensive ac43round and validation o the !ress

model 1ith the air di6user )$OMA+ model.7 $OMA model =

7 Ale to !redict the air5o1 !atterns and thermal distriutions 1ithin a room.

7 8ased on the conservation o mass and ener3#.

7 :et characteristic e0uations 1ere introduced in the model to 3enerali2e its a!!lication to mventilated uildin3s.

7 Sim!lied numerical model and uses !ressure-driven !o1er la1s.



Numerical solution•It is im!ortant to 4no1 e;actl# the numer o e0uations and 4no1n F un4no1

•/or n 2ones in the room, there should e n mass alances and n ener3# alae0uations in the room. 8ut, there are onl# )n-%+ + inde!endent mass alance these n mass alance e0uations, and n inde!endent ener3# alance e0uatio

•ere Zone > is assumed to e adiaatic. This loss is due to 3loal mass consn-2ones.



Mathematical solver

In other 2onal models, to solve non-linear e0uations di6erent metollo1ed,

InardGs $ressure Zonal Model 8ro#denGs method.

8TE8ES model )Hsch1ind+ Hear-Adams method.

9odri3ue2Gs $ressure Zonal model Ne1ton-9a!hson met

OMIS F ONTAM Ne1ton-9a!hson me

$OMA )$ress et al, *(('+ Ne1ton-9a!hson 3loconver3ence method.

Ne1ton-9a!hson method solves the nonlinear !rolem # an iteration o tsolutions o Iinear e0uations.



8ro#denGs method

8ro#denGs al3orithm is an iterative method or solvin3unconstrained non-linear o!timi2ation !rolems.

This method a!!ro;imates the Ne1tonGs method.

It is also 4no1n as uasi-Ne1ton method.

:acoian matri; 1as used at ever# iteration. 8ut is e;!ensi

in all the iteration. The idea ehind 8ro#denJs method is to com!ute the 1holeonl# at the rst iteration, and to do a ran4-one u!date atiterations.



$OMA In $OMA, the Ne1ton - 9a!hson 3loal conver3ence techni0ue 1asto solve the nonlinear e0uation s#stem

Ater that :acoian Matri; 1ill e ormed # the un4no1ns and corresiduals.

8# ne3lectin3 terms o order and hi3her, and # settin3 /);K;+

This 3eneral Ne1ton-9a!hson method 1ill ail, ecause the initial 1onGt e sucientl# close to the root.



onti..

And also ver# slo1 conver3ence as the iterations almost oscet1een t1o di6erent sets o values )@alton %&&<+, so a ac4method 1as selected to nd a !ro!er ste! o variations.

$redictin3 the value or is not more useul. /irst, 1eGve to tr# 1it

value %, this 1ill e sucientl# close to the roots. I, /)P+ doesnGt macce!tance criteria, reduce the value until a suitale !oint is ound

This 1as !ro!osed e $ress et al. )%&&*+



onti..

The criterion or the acce!tale ste! is =

/irst tr#, % and i the ste! is not acce!tale, tr#1here,

8ac4trac4in3,

It ecomes,

ere % and * are the !revious and recent values rom the e0uation.

Different Solver methods used in Zonal Models

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