CALCULATION OFTHEAMOUNTOF ENERGY RELEASED BYHEATING PIPES
INAGREENHOUSEAND ITSALLOCATION BETWEEN CONVECTIONAND RADIATION
Cecilia Stanghellini
ABSTRACT
The energy output from aheating system in agreenhouse ismostly
known onlywithinbroad limitsand forlongperiodsof time.
Thevalues forheattransfer coefficients calculated fora single
pipe in laboratory conditions cannot easilybe applied to conditions
in the greenhouse whereheating pipes are usually grouped within
the canopy.Themethod described allows theheat transfer coefficient
of agivenpipe system tobe estimated using simple temperature data,
measured when the system iscooling.Theparameters of the theoretical
cooling function,i.e. the totalheat transfer coefficient and theapparent thermal conductivity of thepipe system, can thusbe derivedbybestfitting.The smalldifference between the latterparameter and the
thermal conductivity ofwater is discussed.
Itis further shown that,when theeffectof radiation is eliminated
from the totalheat transfer coefficient,theNusselt number can be
calculated for the system.Theresulting relation isthenapplied to
estimate the system's energy output and temperature when cooling.
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CONTENTS
Abstract
•*•
Symbols
*
1 Theory
2 Materialsandmethods
3 Results
4 Conclusions
1"
15
^°
Acknowledgements
19
References
20
SYMBOLS
Capital
Gr =Grashof number =
gl^_
V
T
s -T f
2
H
=energy (J)
Jn
=Bessel function of order
Tf
n
Nu =Nusseltnumber = 1/6
Pr =Prandtlnumber = V / K
Q
=heat flux (Wm - 2 )
Re =Reynolds number = ul/V
T
= temperature (K)
Lower case
r
= correlation coefficient
x
= aspatial coordinate (m)
Italic
a
d
= specific heat (Jkg )
=diameter (m)
g
= acceleration due togravity (9.81ms - 2 )
1
= ageneric dimension (m)
T
= radius (m)
s
= standard deviation
t
= time (s)
u
=airvelocity (ms~l)
Greek
a
=heat transfer coefficient (Wm_2K~l)
6
= thickness (m)
E
= emissivity coefficient
K
= thermal diffusivity = A/pe (m2s_l)
X
= thermal conductivity (Wm^K - *)
v
=kinematic viscosity (m2s"l)
p
=density (kgm~3)
0
= Stefan-Boltzmann constant (5.67 10 - ^ W m - 2 K -
Subscripts
a =air
c =convection
f =fluid
m =mean
r =radiation
s =surface
w =water
5-
CALCULATIONOFTHEAMOUNTOFENERGYRELEASEDBYHEATINGPIPESINA
GREENHOUSEANDITSALLOCATIONBETWEENCONVECTIONANDRADIATION
Toassesstheenergybalanceofagreenhousetheamountofenergyreleased
bytheheatingsystemmustbeknown.Moreover,sincethecanopy(and
thegreenhouseenvironmentingeneral)doesnotnecessarilymakethe
sameuseofradiationandsensibleheat,oneshouldalsobeableto
separatelyevaluatetheamountofenergytransferredintheradiative
andconvectivemoderespectively.
1.THEORY
Ingeneral,heatisbeingtransferredinthreeways:by
convection,
i.e.byairinmotion,and conduction,
radiation,
whichisthe
exchangeofkineticenergyatmolecularlevelinsolidsorstillfluids.
Moreover,twotypesofconvectionareimportant: forced
convection,
ortransferthroughanairstream (therateofheattransferthus
dependsonthevelocityoftheflow);and free convection,
which
dependsonverticaldisplacementofairmassesduetotemperature
gradients.
Inthepresentstudy,radiativeheattransferbetweentwobodiesat
temperaturesTjandT2respectivelywillbeestimatedaccordingtothe
muchusedlinearizationoftheStefan-Boltzmannformula,i.e.:
Qr =ea(Tj1-T|)=4EOT^ (T1 -T 2 )=ar(T1-T 2 )
wm~2 (1)
wheretemperaturesareinKandthesubscriptmdenotesthemeanofthe
twotemperatures.
Ontheotherhand,conductiveheattransferthroughanairmassin
agreenhouseenvironmentcanbeneglectedduetothesmallconductivity
ofstillair (A =2.53-10-2Wm _1 K _1 ,fordryairat15°C)whichis
manyordersofmagnitudesmallerthanan"apparentconductivity"for
convectionevenatthefairlylowairvelocitiesexperiencedina
greenhouseenvironment.
Heatfluxduetoeonvectioncanbedescribedformallybytheequation
forconductionofheatinasolid:
Q=-XVT
Wm~2 (2)
Whensuchanequationismeantforheattransferthroughamovingfluid,
Awillratherbean"apparentconductivity"andthethicknessofthe
layerwherethetemperaturegradienthastobeconsideredwillbe
an"equivalentthickness",accountingfortheneteffectofthe
presenceofasurfacelosingheattothemovingfluid.
Non-dimensionalgroups,suchasthevariousnumbers,helpinsolving
particulartransferproblems.Thus,theGrashofnumber (Gr)isa
measureoftheimportanceofbuoyancyduetothedifferenceintemperature
betweenthesurfaceandthefluid,whiletheReynoldsnumber(Re)
describestheeffectofthesurfaceitselfonturbulenceinthefluid.
AtalargeReynoldsnumber,heattransferisduemoretothefluidstream
thantobuoyancyandforcedconvectiontakesplace,whilewithlarge
Grashofnumbers (Gr>Re2)thereverseistrue,andtheprevailing
transfermechanismisfreeconvection.
Inagreenhouseclimateitisoftenacaseofmixedconvection,since
itisusuallyimpossibletospecifytheprevailingdrivingforcefor
heattransfer.Anothernon-dimensionalgroup,theNusseltnumber(Nu),
canbehelpful,beingtheratiobetweenacharacteristicdimensionof
thesurfacetothethicknessofan"equivalentboundarylayer"through
whichthetransferofheattakesplace.TheNusseltnumberisafunction
eitheroftheGrashofnumberinfreeconvectionoroftheReynolds
numberinforcedconvection.
IntheintermediateregionitisagoodpracticetocalculateNuas
afunctionofeitherandusethelargestNutoestimatetherateof
heattransfer.Theheattransfercoefficientforconvectiona is
c
easilyrelatedtoNu.Infact (2)canbewrittenforone-dimensional
convectionthroughtheequivalentboundarylayeroffinitethickness6
(Tg-Tf)
_
Q c=X
'
"m * (3)
6
whereT^denotestheprevailingtemperatureofthefluidatdistance6
fromthesurface,andthesignispositiveforfluxesleavingthe
surface.AftersubstitutionofNu= 1/6, (3)becomes
A
Q c =Nu- (Ts -T f )=Ctc (Ts -T f )
-2
Wm
(4)
where îisatypical dimension of the surface.
Since radiative heat loss isalsopresentwhenadifference in
temperature isexperienced,a surface immersed ina fluid losesheat
according to
Q = (<xr+a c )( T S -T f )=a(T s -T f )
Wm~
(5)
where atotalheat transfer coefficienta isdefined as the sum of
the radiative and convective coefficients.
Ifit isassumed that theenergy input inagreenhouse by radiation
of theheating system isknownwhen the temperature of the heating
elements isknown (eq,(l)),and conduction isneglected,the problem
ofestimating the energy released by theheating system is reduced
totheknowledge of a
First,ithas tobe decided whether theprevailing transfer mechanism
is freeor forced convection,according to themethodbriefly described
above.
Forheatingpipes thediameter isanatural choice for the typical
length; then forpipes ofdiameter d =0.06 m, 15°cwarmer than the
—1
surrounding air,and anairvelocity u s0.1 ms ,Re =4*10 and
c
_5 o-1
2
Gr =4*10 (kinematic viscosity ofairv = 1.5-10 m s ) .It appears
then that free convection should bemore important than forced convection
andNu should be calculated asa functionofGr.
There isagreement in the literature that for systemswith Gr <10
Nu=C(Pr-Gr)1 / 4
g
(6)
with C a constant and Pr thePrandtl number of the fluid (0.71 forair).
The value of the constant depends on the shapeof the surface losing
heat, andmostly there isno complete agreement on it:Jodlbauer (1933)
proposed C =0.455 for longhorizontal cylinders,whileMcAdams (1954)
suggested avalue of 0.525.Monteith (1975)quotes C =0.523 from
previousworks and Stoffers (1976)calculated C= 0.614 from measurements
inagreenhouse environment.
Since,assubstitutionof (6)in (4)shows,
a c=CA (Gr-Pr)1/4
Wm~2K-1 (7)
aninaccuracyof30%,say,inCmeansthesameinaccuracyinthe
convectiveheatflux;itappearsthenthatagoodestimateofCfor
agivensystemisthebottlenecktotheknowledgeofitsenergy
output.
-9
2.MATERIALSAND METHODS
As stated above,anestimate of theenergy input from theheating system
andof its allocation between convection and radiation,wasamuch
neededpartof a larger experiment aimed atdefining the energy balance
of agreenhouse.The experiment isdescribed elsewhere:Stanghellini
(1981); Bot et al. (1983);van 'tOoster (1983); Stanghellini (1983),and
only the relevant instrumental set-up for thepresent subjectwill
be described here.
The greenhouse isaVenlo-type,8spans 3.20 m each,22m long,
oriented E-W.Heating isprovided byhotwater circulated inpipes
(diameter: 5.735'10
m) laid a few centimeters above ground,one
element (supply and return lines) foreach canopy row.A secondary
system isatgutter level,onepipe foreach span.Toprevent side
effects,a setof fourpipes one above theother isbrought to each
wall.Thewall network isdirectly coupled totheground one,while
the gutter system is switched inonly ifnecessary.The total length
of theground network is588m, thatof the gutter one 164m,while
thewall system is 433m in length.Thisadds up toatotal volume
ofwater inthepipe system of 3.06 m .Thewater inthepipe system
isheated inaheatexchanger inwhich steam is circulated.Abuffervessel (0.4m )isplaced inparallelbetween theheatexchanger and thepipe
system.Water is continuously circulated in thepipe system,evenwhen
the valvebetween itand theheatexchanger-vessel system is closed.
The total energy input tothe greenhouse wasmeasured in twoways:
a.bymeasuring the amount and temperature of condensed water
flowing atthe outlet of theheatexchanger in the greenhouse;
b. bymeasuring thewater flow tothepipe system and temperature
decaybetween the inletand outletof that system.
It soonappeared thatenergy inputestimateswith the firstmethod
wereby fartoohigh, leading to the conclusion that some steam is
being condensed already at the inletof theheatexchanger;no further
attemptwasmade to improve the reliability of thismethod.For the
second one,on theotherhand,a commercial devicewas installed,which
workedproperly for about threemonths. Itceasedworking for reasons
- 10
stillunclear;itis,however,possiblethatafaultintheheating
system,whenboilingwaterwaspushedintothepipes,hasdamaged
thesensoroftheflowmeter.Thetemperatureofthegroundpipeline
systemwasmeasuredbytwothermocouples (onthesupplyandreturnlines)
atonesectioninthemiddleofthehouse;thatofthegutteronewas
measuredinthesameway,ataboutthesamespot,whilethetemperature
ofthewallsystemwasnotchecked.
ThetemperatureoftheairwasmeasuredbyAssmannaspiratedpsychrometers
atvariousheightsaboveground.Airtemperaturereferredtohereafter
wasmeasured0.8mabovethepipes.
Theairvelocitywasmeasuredbyfourhot-bulbanemometers (denOuden,
1958)alsoatvariousheightsabovethepipesystem.Hereafterthe
averageofthefourvalueswillbeusedforeachmeasurement.
Thetotalheattransferfromthepipesystemcouldbecalculated
whilewarmingitupandthenlettingitcooldown.Infact,the
temperature (T)asafunctionoftime (£)anddistancefromtheaxis(x)
ofacircularcylinderofradius (r)andinfinitelength,losingheat
atitssurfacetoamediumatzerotemperatureaccordingtoalinear
law,aseg. (5),isknown (CarslawandJaeger,1948)
_ß2 < t
n
To
r2
w=l
J0<25a>
2A
3^+A2
r
J0(ß„)
whereT 0istheconstanttemperatureofthecylinderattime t =0;
&Xaretherootsofthetranscedentequation
&Jl(B)=AJ0(3)
(9)
andAisrelatedtoaby:
A=£ a =r_°U
(10)
i.e.excanbecalculatedfromAand<iftheyareknown.
Thiswasachievedbydefininga"meanpipe"withasurfacetemperature
whichisthemeanofthemeasuredones,andthenfindingbytheleast
squaremethodthebestfitof (8)onthecoolingportionofthe
temperaturevs.timefunction.Itshouldbenotedthatwhentemperature
ismeasuredatadistancexfromthepipeaxis,equaltoitsradius r}
11
Table 1-Averagevaluesovereachtimeinterval,onwhichpipe
surfacewascooling,usedforthebestfitprocedure.
Symbolsareasusedinthetext,temperaturesarein°C
andtimesinhoursandminutes.
Timeinterval
Date
from
to
Feb.20,'81
18.10
19.40
20.10
Feb.21,'81
s
T -T
s a
T
m
14.80
42.85
28.05
28.83
T
a
T
21.30
14.64
44.35
29.70
29.50
22.00
23.30
14.39
45.49
31.11
29.94
0.00
1.20
14.30
45.64
31.34
29.97
1.50
3.10
14.13
46.23
32.11
30.18
3.40
5.00
14.60
45.90
31.30
30.25
5.30
6.50
14.49
46.30
31.81
30.39
7.20
8.10
16.04
52.28
36.24
34.16
Mar.14,'81
19.40
23.40
14.07
28.11
14.04
21.09
Mar.15,'81
0.00
3.50
14.33
28.67
14.34
21.50
4.20
7.20
15.31
32.75
17.44
24.30
Apr.15,'82
22.44
1.26
17.14
33.13
16.00
25.13
Apr.16,'82
1.56
4.23
17.06
34.64
17.58
25.85
II
- 12-
thelastfactorontheright-handsideof (8)equalsone.Moreover,
onlythefirsttermofthesummationwasfitted.
Sincethesurroundingaircouldnotbeconsideredatconstanttemperature,
thedifferencebetweenthemeasuredpipeandairtemperaturewasused
foreachstep.Anyhow,inordertodecreaseinaccuracyduetothis,
thecalculationsweremadeonlyonnight-timevalues,thusminimizing
airtemperaturevariationsnotduetopipeheating.
Calculationswereperformedon13coolingperiodsofthreenights
withvariousclimaticconditions.Averagevaluesforeachcoolingare
summarizedinTableI.Thestrictrelationshipshouldbenotedbetween
airvelocityandpipe-airtemperaturedifference,showninFig.1
forthesamedata.Thisisactuallyanindicationthatmostoftheair
movementisduetobuoyancy.Itappearsthatafunctionofair
velocitycouldaswellbesubstitutedineq. (7)insteadofthefunction
ofthedifferenceintemperature,leadingtothesuggestionthatin
suchconditionsNucouldalsobecalculatedasafunctionofthe
Reynoldsnumber.Inanycase,thisismoreanacademicquestionthana
practicalone,sinceitiseasierandcheapertomeasuretemperatures
thanairvelocitiesatsuchlowlevels.
Theregressioncoefficientscalculatedfortheprocedureoutlined
abovewerealwaysabover=0.99andnoinfluenceofthetimeinterval
betweensubsequentdatawasrevealed;dataeverythreeortenminutes
wereused.Themagnitudeofthe2ndordertermin (8)wascalculated
a posteriori:
itscontributionwasabout0.5°Cinthefirstfewsteps
andbecamequitenegligibleafterafewminutes.
13
o
4
•
S
Ê
O
>tH
O
O
_1
iü
>
H
<
UJ
CD
<
OL
UJ
>
<
25
38
35
AV.PIPE-AIR TEMP.DIF"C
Fig. 1 - Air velocity as a function of the difference
in temperature between the pipe surface and
air 0.8 m above ground. Each point shows
averages over a whole cooling interval.
- 14
3.RESULTS
Thecontinuouscirculationofwaterinthepipesystemintroducesa
perturbationinthecoolingpattern.Duetotheforcedcirculation,
itmustbeassumedthatconvectionandnotconductionisthemainheat
transfermechanism,alsoinsidethepipe:ithastobeexpectedthat
temperaturedecayofthepipesurfacepointstoahigherconductivity
(andhencediffusivity)thanifstillwaterwaspresent.Ontheother
hand,itisacknowledgedthattheheatexchanger-buffervesselsystem
isalwaysgainingheatduetoleakagesthroughthevalveplacedatthe
steaminletoftheheatexchanger.Thevalvebetweentheheatexchangervesselloopandthepipeloopisalsoleaking,althoughtoalower
degree;itmaybeconcludedthenthatthewatersteadilybroughtto
anygivensectionofthepipesystemiswarmerthanthewaterit
replaces,i.e.thecoolingrate (anddiffusivitydirectlyrelatedtoit)
issmallerthanitshouldbe.
Theseconsiderationsandthepresenceofmanydifferentthermalconductivitiesbetweentheaxisofthepipeandthesurfacewherethe
temperatureismeasured (water,iron,coating)leadtotheconclusion
thatdiffusivityKhastobeanoutputparameterofthebest-fit
procedureandnotaninput,i.e.anapparentdiffusivityforsuchasystem
iscalculated.
ThecalculatedKastheaverageoftheresultingdiffusivitiesforeach
coolingintervalis1.12*10 mzs7 9-1
(withastandarddeviation
s=0.314*10")whichisindeedsmallerthanthediffusivityofwater
K =1.52"10~ ms" at40°C.Itmeansthattheeffectofleakages
fromtheheatexchangeroutweighstheeffectofconvectioninsidethe
pipesection,andthatofthemuchgreaterdiffusivityofapartofit
-5 2-1
(K. =1.2-10 ms ).
iron
Usingthetotalheattransfercoefficientacalculatedfrom (10)the
energyHreleasedbythepipesystemwascalculatedastheintegral
ofÖoverpipeareaandtimeforthetwonightsforwhichenergy
consumptiondataofthegreenhousewereavailable.Thetotalamount
ofheatreleasedduringthe8coolingintervalsofFebruary20-21,1981,
g
wasH=2.81*10 j (theguttersystemwascoupledtothemainnetwork)
andthetimeintervaloverwhichreadingsoftheenergymeterare
- 15-
50
45
48
û
Lü
>
a.
35
Lu
</>
m
o
Lü
a
<
ce
LU
CL
Lü
LU
CL
H
CL
2
4
27-28MAR1981
Fig. 2-Observed and calculated pipe surface
temperature (usingeq. (8)).The starting
temperature for each cooling is input into
theprocedure.Note thatwhen the air
temperature is influenced by other factors
thanpipeheating (after 8.30 a.m.) the
predicted temperature deviates more from
themeasured one.
- 16
hours
available is 33% longer (some time and one more warming-cooling after
the last temperature measurement,plus all the intervals when the pipe
Q
temperature was rising); thispoints to a value H = 3.72*10 J for the
9
whole time,which is in good agreement with H = 3.96*10 J provided
by themeter. It is suggested that some energy is lost outside the
heating system, in the connecting sections.On the other hand, for the
measurements of 14-15March, 1981,the time interval between readings
of the meter was 10%longer than the total cooling time,and the
gutter system was not connected. The total heat released in the three
cooling intervals was H = 1.01*10 J x 1.1 = l.ll'lO 9 J that is
g
again a small underestimate of H = 1.16*10 J provided by the meter.
In order to enable estimates of heat released in other circumstances,
without having to go through this cumbersome procedure all over again,
c*cwas calculated from each a by means of (5) (a r was calculated from
(1)with E= 0.95),and the corresponding Nusselt number according
to (4)was written,using the above-mentioned value for X
it was
then calculated as a function of the Prandtl and Grashof numbers,
i.e. the value of the constant C of (6)was estimated. Itwas found
that for such a system
Nu = 0.330 (Gr.Pr) 1 / 4
(11)
with a standard deviation of C,s = 0.048.
The value of the coefficient is lower than any value found for similar
systems (with stillwater,however) in the literature cited,and can be
explained by the considerations made above.
Eq. (1),(4)and (11)were substitutecKin (5)to get
1/4
a = 2.155-10" 7 T 3 + 3.505 t—B
m
2_)
Wm~ 2 K~ 1 (12)
x T
T a . .d d '
and (8)with substitution of (12)was applied to data of a fourth
night, to calculate the pipe temperatures.The results are shown in
Fig. 2,together withmeasured values and allow the conclusion that
(12) gives satisfactory estimates of the convective (and radiative)
heat transfer coefficient for such a system.
17
4.CONCLUSIONS
Theheattransfercoefficientthuscalculatedforconvectionaveraged
65%ofthatforradiation.Thisshowsthatforsuchasystemradiation
isamoreefficientwayoflosingenergythanconvection.From
literaturevalues,suchastheonesmentioned,itismostlyconcluded
thatinthepresentrangeoftemperatures,suchasystemshouldlose
abouthalfofitsenergyeitherway,whilethefindingsofStoffers
(1976)indicatedthatconvectionshouldbemoreefficient.Ithas
tobestressedthattheconditionsofthevariousexperimentswere
hardlysimilar,andnoattemptismadeheretoinvestigatethe
combinedeffectofthemanydifferences.
However,itmaybeconcludedthatthemethodoutlinedhereallows
agoodestimateoftheheattransfercoefficientstobemadefora
givensystem,bymeansofafewandsimplemeasurements.Ithastobe
stressedthatnoactionhadtobeundertakenonthesystemtomake
suchcalculationspossible.Inthisway,theenergyoutputfroma
givensystemunderitsnormalworkingconditionthroughbothconvection
andradiationcanbeeasilyestimated.
- 18-
ACKNOWLEDGEMENTS
FinancialsupportfromtheDutchMinistryofAgricultureforthe
wholeprojectisgratefullyacknowledged.Iamverymuchindebted
toIr.W.P.Mulderandthestaffofthe"ClimateControl"department
ofIMAGfortheirstimulatingsupport.Ir.G.P.A.Botofthe
AgriculturalUniversityinWageningencontributedheavilytothe
wholeresearchandF.J.S.M.Dormans,A.van 'tOosterandJ.E.M.Reinders,
studentsofthesameUniversity,putintoitalotoftheirskill
andenthusiasm.DiscussionsaboutthissubjectwithDr.M.Menenti
andDipl.Phys.J.A.Stofferswereformealwaysfruitful.
19
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DasTemperatur und Geschwindigkeitsfeld imgeheizten Rohr bei
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Heat Transmission.McGraw-Hill,New York:pp 172-176.
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Principles of Environmental Physics.
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Metingen aan en simulatie van het klimaat in een geschermde kas.
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Evaporation and energy consumption ingreenhouses.
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Stanghellini,C M . , 1983
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Heat transfer measurements in screened greenhouses.
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20-