Mathematical modeling on the air indoor exchange.docx

Mathematical modeling on the air indoor exchange.docx

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Mathematicalmodelingontheairindoorexchange

Mathematicalmodelingontheairindoorexchange

LinkeXiao

YongkangRen

ZhongyuRen

Chongqingunniversityofscienceandtechnology

Unniversitytown,ChongqingCity

Abstract

It’scoldinwinterinthenorthernChina,andthetemperatureindoorandoutdoordiffersmuch.Tokeepboththeairinsidefreshandthehousewarm,weneedtocontemplatehowlongandhowoftenweopenwindowsanddoors.WetaketheclimateofJinanCityasasampleanddesignamathematicalmodeltoinvestigatetheissue.Underthegivenconditionthattheamountofthepeopleindoorisinvariable,wesetanopen-windowsanddoorsstandard,whichis,whetherthedensityofcarbondioxideandthetemperatureindoormayposethepeoplesick.Combiningphysiologicaldataofhuman,seamsofwindowsanddoorsandflowvelocityofair,wedevisemathematicalmodelstoinvestigatehowthedensityofcarbondioxideindoorchangesduringthedaytimeandnight.Thenconsideringheatchangeswhenopeningwindowsanddoors,wedeviseanothermodeltoinvestigateheatchangesovertime,utilizingMatlabtodescribeit.Finally,wecompromisethefactorsaboveandobtainanoptimalconfiguration.

Keywords:

airindoorexchange,heatexchange,mathematicalmodeling

Introduction

Question:

InthewinterofthenorthernChina,thetemperatureindoorandoutdoordiffersmuchbecauseofthechillweather.Andonlythetemperatureindoorkeptabove18℃canpeoplefeelsuave.Hencepeoplethereusedtoclosewindowsanddoorstokeepwarm.Meanwhileairexchangedecreaseswithwindowsanddoorsclose,whichleadstoworseairconditionandincreasestheriskpeoplecontractrespiratorydisease.“Oftenopenwindowsanddoors,letfreshairin”ThisconcepthasarousedgreatconcernamongpeopleinnorthernChina.Therefore,withagivenhousescaleandknowntemperaturesindoorandoutdoor,weneedtoresearchhowoftenandhowlongshouldpeopleopenwindowsanddoors.Assumethatthehouseis2.7metershigh,3peopleliveindoor.Pleasedesignmathematicalmodelstotackletheissuesbelow.

1.Discusstheconditionsthathousescaleisrespectivelycontrolledin20m2,40m2,60m2and80m2.

2.Thehousescaleneededatleastwithwindowsanddoorsclosethewholewinter.

3.Anyotheraspectsyoushouldtakeintoconsiderationwhilediscussingtheairconditionindoor.

Andtotackletheseissues,weneedtomakecleartheunknownsbelow.

1.Howtheconcentrationofcarbondioxideindoorchangesovertime.

2.Howheatchangesovertimewithwindowsanddoorsopen.

3.Thefrequencyandthelengthofthetimepeopleopenwindowsanddoors.

Analysisofissue

BecauseofthefreezingweatherinwinterinnorthernChina,peoplethereusedtoclosewindowsanddoorstokeepwarm.Thusitdeclinestheairexchange,posingaterribleimpactonpeople’shealth.Thisproblem,whichiscalledindoorairpollution,isarousedbyarchitecturematerials,commodities,chemicalsandhumanactivities.Normally,endogenouschemicalcontaminants（suchascarbondioxide,ammoniaandhydrogensulfide）releasedfromhumanbreath,sweatandurine,exogenouspollutants（suchasbenzene,methylbenzene,styrene,methylalcohol）releasedfromhumanbreath,andvirusandbacteriafromcoughandsneezeconstitutethemajorsourceofthepollution.Althoughindoorairpollutionconsistsofseveralfactors,weusedtoestimateairconditionbyoneofthem.

Carbondioxideindoormostlyoriginatesfromhuman’sbreath.Whileanadultisresting,thecarbondioxidehebreathesoutisabout20L.Andwhileheislaboring,thecarbondioxidehebreathesoutis0.5~1timesmore.Withthelaborloadincreasing,therestofhisbodyinitiatestodischargethepollutionair（suchasdecompositionproductfromsweat.）Andtheconcentrationofoxygenisdecreasingatthetimetheconcentrationofcarbondioxideisincreasing.Whentheconcentrationofcarbondioxidegoesabove0.07%,peoplestarttofeelsick.Sowecanutilizetheconcentrationofcarbondioxideasanindoorairpollutionstandard.Theconcentrationofcarbondioxideoughttoberestrictedunder0.07%.Althoughseamsofwindowsanddoorscanalsocompleteairexchange,itdoesn’taffectsignificantlywhenthedegreeofthecarbondioxidefrombreathishighandwhenthehouseisenormous.Thus,weneedtotakehumanfactorintoaccountwhiledeviseanairexchangemodel.

Toinvestigatetheissue,weshouldreconcileboththeconcentrationofcarbondioxideandthetemperatureindoor.Westipulatethattheconcentrationofcarbondioxideindoorshouldn’trunsabove0.07%,orpeoplewillopenwindowsanddoors.Withwindowsanddoorsopen,heatinitiatesexchanging,coolingdowntheindoortemperaturebymeansofheatconvection.Henceweconsidertheappropriatenessofthetemperatureindooratthesametime.Ifthehousescaleislargeenough,therewillbeenoughseamstocontrolthedensityofcarbondioxidewithin0.07%.Thereforepeopletheredon’tneedtoopenwindowsanddoorstoexchangeair.

Assumptions

Note:

themodelsweusearebasedontheclimateofJinanCity.

Herearetheparameters:

Name

Value

Averagewindvelocityoutdoor

3.2m/s

Averagewindvelocityofmajordirection

4.3m/s

Majordirection

ENE

Incidenceofmajordirection

15%

Averagealtitude

51.6m

Temperatureoutdoor

-7℃

DensityofCO2outdoor

0.588g/m3

Name

Value

ConcentrationofCO2outdoor

0.039%

LimitoftheconcentrationofCO2

0.07%

QuantityofflowonCO2frombreath

40g/h

QuantityofflowonCO2whilesleeping

25g/h

Densityofairoutdoor（18℃）

1.213kg/m3

Densityofairoutdoor（-7℃）

1.327kg/m3

Specificheatcapacityoutdoor

1.009kJ/（kg.℃）

Specificheatcapacityindoor

1.013kJ/（kg.℃）

Assumptionsonthehousestructure:

Monolayerirondoor,size（width*height）:

1.5*2m,inevitableseam:

9m

Monolayerglasswindow,size（width*height）:

0.5*1m,inevitableseam:

8m

Flowquantityofairosmosisofdoorseam（eachmeter）:

2.86m3/（m*h）

Flowquantityofairosmosisofwindowseam（eachmeter）:

1.98m3/（m*h）

Proofcannotpreserveheat.

Herearetheillustrationsofwindowanddoor:

Otherassumptions:

1.Withwindowsanddoorsopen,wedimtheairflowisone-dimensionalsteady.

2.Thetemperatureindooriscontrolledat18℃withwindowsanddoorsclose.

3.Weassumeonemorewindowevery20m2gainedofthehousescale.

4.Weassumeonlycarbondioxideaffectstheaircondition.

5.Weassumethecircumstanceconditionsoutside（suchastemperatureoutside,frequencyofwind,flowdirection）aresteady.

6.Theairhasbeenmingledinstantlywhenexchanged.

7.Ignoreseamswithwindowsanddoorsclosed.

8.Ignoreresistancewhenairisflowing.

9.Theairisuniformdistributedandairflowsonlyaroundexitsandentrances.

10.Thetreepeopleareafamily,whichcontainstwoadultsandakid.Andthegasakidbreathishalfthatofanadult.

Symboldefinition

Symbol

Definition

ρ1

Airdensityoutdoor

ρ2

Initiateairdensityindoor

L1

Flowquantityofairosmosisofdoorseam

L2

Flowquantityofairosmosisofwindowseam

V1

Housevolume

Vq

Flowquantity

S

Housescale

Symbol

Definition

v1

Flowvelocity

Ⅱ1

Doorseamscale

Ⅱ2

Windowseamscale

c1

specificheatcapacityoutdoor

c2

specificheatcapacityindoor

T1

Initiatetemperatureindoor

T2

Temperatureoutdoor

T3

Limittemperatureindoor

Sm

Doorscale

Sc

Windowscale

A

Breathquantity

A’

Sleepingbreathquantity

Cco2

Concentrationofcarbondioxideoutdoor

C’co2

Concentrationofcarbondioxideofsickpoint

C’’co2

Limitconcentrationofcarbondioxide

Model1

Weinitiatetoinvestigatethechangeofcarbondioxideovertime,regardlessoftemperaturelimit.

Wedetectthatpeopleshouldopenwindowsanddoorsinsuchcircumstance:

（1）

（2）

Intheinequationabove,

denotesthetimechargedtodeclinetheconcentrationofcarbondioxide.Attheleftsideinparentheses,thefirsttermrepresentsthedensityofcarbondioxideindoorwhenpeoplebreathe.Thesecondtermrepresentsthedensitychangeoftheosmosisquantityfromseams.Thosetwoterms’differencemultipliedbytheflowquantityrepresentsthesumquantityofcarbondioxideindoor.Whenthevaluegoesabovethelimitthatmakespeoplesick,peopleshouldopenwindowsanddoors.

Model2

Basedonmodel1,weinitiatetoconsiderthetemperaturefactor.

Consideringthattheflow-inquantityvaluesfromabouteighty-percenttoninety-percentoftheflow-outquantityinreality,weassumethattheproportioniseighty-fivepercent.Andhereisthemassofcarbondioxidewhenopenwindowsanddoors.

M1=M1+

-（M1+

）/V1*v1*Sc*dt+1.372*0.039*0.01*v1*Sc*dt*0.85.

M1=M1+

-（M1+

）/V1*1.6dt+1.372*0.039*0.01*1.6dt*0.85.（3）

Heatkeepstransferringthroughdifferentsystemsuntilthetemperaturereachesequality.Thewholeprocessobeystheenergyconservationlawandtheenergytransferlaw.Energytransfersfromthehigh-temperaturesubstancetothelow-temperaturesubstance.Virtuallyit’satransferofinternalenergy,thatis,theabsorptionoftheinternalenergyfromthelow-temperaturesubstanceisequaltothereleaseoftheinternalenergyfromthehigh-temperaturesubstance.

Thethermalbalanceequation（Q1=Q2）onlyworksintheprocessinadiabaticsystem,thatis,thewholesystemwithnoenergyloss.Andmoreover,theinitialstateandthefinalstateshouldbeequilibriumstate.Usually,thereleaseofcalorieresultsfrominclineoftemperature,solidification,liquidationandburning.Andonthecontrary,theabsorptionofcalorieresultsfromincreaseoftemperature,dissolutionandevaporation.

Accordingtothethermalbalancelaw,wecomeupwiththeequationbelow:

（4）

WhereVqisequaltov1multipliedS’,and

i