2.0 ANALYTICAL MODELS OF THERMAL EXCHANGES IN THE PYRANOMETER

Transcription

1 2.0 ANAYTICA MODE OF THERMA EXCHANGE IN THE PYRANOMETER In Chapter 1, it wa etablihe that a better unertaning of the thermal exchange within the intrument i neceary to efine the quantitie proucing an offet. In thi chapter, we review everal raiative tuie of the pyranometer an a imilar raiometer, the pyrgeometer. The concluion of thi chapter i a correction moel for the intrument meaurement. 2.1 Pyrgeometer Moel everal analytical moel have been evelope for the pyranometer iter intrument, the pyrgeometer. In 1970, Drummon et al. ecribe the evelopment of an intrument eigne to irectly meaure longwave raiative fluxe. Ue of thi pyrgeometer replace the proceure of etermining longwave fluxe by ubtracting the meaure hortwave from net fluxe [Albrect an Cox, 1977]. The Eppley Preciion Infrare Raiometer (PIR) moel pyrgeometer ue the ame thermopile principle of operation a the pyranometer to meaure incient longwave raiation (-50 µm). The enor of the PIR i hiele by a filter that limit the reponivity of the intrument to the thermal pectrum. It ha been hown that the KR-5 ome with an interference filter eigne to tranmit thermal raiation will aborb an reflect olar raiation in the viible pectrum [Duchon an Wilk, 199]. Thi create a temperature ifferential between the ome an the enor. A in the cae of the PP, thermal raiation emitte from the interior of the ome a to that tranmitte through the ome, offetting the true meaurement ignal. For everal year, variou group have attempte to quantify thi varying offet ue to olar heating an thermal raiative cooling in orer to evelop an analytical correction for the pyrgeometer ignal. In 1977, Albrecht an Cox evelope a proceure to improve meaurement of the pyrgeometer by coniering the thermal exchange within the intrument. An analytical expreion for a buget of energy within the intrument reveale that the actual flux being meaure from a ignal output coul be repreente by 3 ( c c T ) σt k ( T T ) F = σ, (2.1) where F = incient irraiance to the intrument (W/m 2 ), U emf = thermopile ignal (V), 11

2 Amie M. mith Chapter 2. Analytical Moel of Thermal Exchange in the Pyranometer 12 T = temperature of enor (K), T = temperature of KR-5 filter (K), o = effective emiivity of the enor urface, an σ = tefan-boltzmann contant = 5.67 x 10-8 W/m 2 K. The manufacturer calibration of the pyrgeometer etermine the contant of k, c 1 an c 2. Internal circuitry i eigne to compenate for the T 3 an T term. The (T -T ) term oe not come into play becaue the ome an enor are maintaine at the ame temperature uring calibration. During operation, however, the olar heating typically prouce a much a a 10 C ifference between the filter ome an the intrument boy temperature. Uing their analyi, Albrecht an Cox etermine that a 0.1 C uncertainty between ome an heat ink temperature may reult in a 3 to W/m 2 error in meaurement, which make the olar heating effect ignificant. They propoe moifying the intrument to monitor of the ink an ome temperature a a function of time, an then to ue that information with the thermopile output to increae the accuracy of the ata. Philipona et al. [1995] propoe a moification to the pyrgeometer that provie the temperature information neee to account for thermal fluxe within the ome. A imilar energy balance wa performe within the KR-5 ome. The Albrecht an Cox equation, Equation 2.1, wa moifie lightly to inclue higher-orer term an previouly etermine intrument quantitie. The propoe correction to be applie to the pyrgeometer ignal i E 3 ( k T ) k σt k ( T T ) = 1 1σ B 2 B 3σ, (2.2) D c B where c = enitivity of the pyrgeometer (V / Wm -2 ), an k 1, k 2 an k 3 are parameter that may be erive from the optical propertie of the ome an enor an the operational characteritic of the thermopile. The analyi by Philipona et al. oe not treat multiple reflection within the KR-5 ome. In fact, multiple reflection of thermal raiation are not much of an iue in the pyrgeometer ince the filter are eigne to tranmit thi raiation. That i, thermal raiation leaving the enor urface, or other internal urface, i much more likely to pa through the filter ome out of the intrument than to be reflecte back on the enor an interfere with the

3 Amie M. mith Chapter 2. Analytical Moel of Thermal Exchange in the Pyranometer 13 ignal. In the pyranometer, however, the filter are eigne to aborb or reflect thermal raiation o that multiple reflection among the internal urface mut be coniere. Philipona et al. propoe to moify the pyrgeometer by aing three thermitor to the inie urface of the ome at a 5-eg angle from the bae an pace 120-eg from each other. The average of thee three thermitor woul repreent any axial variation in ome temperature. The 5-eg location wa hypotheize to repreent variation in ome temperature along the ome zenith. 2.2 The Buh Moel Algorithm to correct for the zero offet in the pyranometer have been evelope uing experimental reult. In 1998, Buh et al. ecribe an effort to correct pyranometer meaurement uing an empirical moel to account for thermal exchange between the ome an the enor. The reult of thi moel i a linear approximation to the ifference of the temperature to the fourth power. The correction to be applie in thi moel i expree a the offet quantity 8 F = ( T Dome T Detector ) [W/m 2 ]. (2.3) The coefficient in Equation 2.3 apply to the intrument ue in the experiment. Thee woul be etermine for an iniviual intrument in calibration. In operation, thi correction require that the temperature of the ome an enor be continuouly monitore to correct ata for a correponing perio of time. Buh et al. propoe a moification to the pyranometer that employe four thermitor. Two thermitor are mounte uner the intrument floor near the enor an two are attache to the outer ome. The ual thermitor ample patial variation in temperature. Thi moel i trictly empirical an oe not involve the phyical procee of energy exchange within the intrument. pecifically, it neglect the aitional exchange within the intrument ue to reflection an wavelength-epenant urface propertie in favor of a linear approximation. The effect of thee trategic moification on the performance of the pyranometer mut alo be a ource of concern. The temperature of the enor cannot be irectly meaure without altering it temperature, thereby affecting the meaurement. Therefore, the team ecie to compromie an place the thermitor near the enor rather than irectly on the enor to obtain an approximation to it temperature. The hea of the thermitor mounte on the ome

4 Amie M. mith Chapter 2. Analytical Moel of Thermal Exchange in the Pyranometer 1 were hiele from irect unlight o that they woul not aborb olar raiation. However, thi hieling, a well a the thermitor an lea will block ome olar raiation from reaching the enor. The hieling will alo reuce the local temperature on the ome itelf. 2.3 The Haeffelin Moel In a 1999 paper, Haeffelin et al. propoe a correction for the pyranometer imilar to that emboie in the Philipona moel of the pyrgeometer. They conclue that the intrument meaurement may be correcte to the eire hortwave irraiance, E, by 1 σ 3 σ σ E = (1 ρ ) T ( T T ) (1 ) T b α b b α c. (2.) The phyic behin the correction emboie in Equation 2. are ecribe in etail in ection Phyic within the Haeffelin Moel An analyi imilar to that of Philipona et al. ecribe the phyical procee involve in the thermal offet. Reaonable aumption are ue to arrive at Haeffelin correction expreion, Equation 2.. The origin of the net raiative flux arriving at the enor urface are etermine uing the raiation moel hown in Figure 2.1. Three ource of raiation are coniere. The incient olar flux on the intrument arrive after being tranmitte through the filter. Thermal raiation emitte from the ome arrive at the enor urface. In aition, thermal raiation emitte by the enor urface i reflecte back to the enor by the ome, which oe not allow thermal raiation to tranmit. Thi latter raiation either return to the enor urface, or i aborbe by the ome, thereby contributing to the ome heating.

5 Amie M. mith Chapter 2. Analytical Moel of Thermal Exchange in the Pyranometer 15 Figure 2.1 Raiative heat tranfer in pyranometer analytical moel Though the hortwave an longwave component of flux arriving at the enor urface may be foun eparately, the enor meaure total net flux. The optical propertie of the lacquer are eentially black for all wavelength, o that the longwave ( ) an hortwave ( ) propertie are the ame. Thi lea to the relation F net = E α, (2.5) F net = E α σt ( φ) α σt ρ α σt, (2.6) an F net = α E α σt α ρ σt σt ; (2.7) where E = external irraiance (W/m 2 ), longwave irraiance i blocke by ome, an T = inner ome temperature, it i aume a repreentative ome temperature can imulate the flux conition of the ome graient, T (φ). The ignal prouce by the thermopile i aume to be a linear function of the ifference in temperature between the enor urface an the intrument boy temperature when operating uner normal conition. The ignal may be converte to an irraiance meaurement uing the enitivity relation

6 Amie M. mith Chapter 2. Analytical Moel of Thermal Exchange in the Pyranometer 16 U = T T ) = c( F ), (2.8) emf ( b net where U emf i the voltage ignal (µv), i the thermopile eebeck coefficient (µv/k), T an T b are the enor an boy temperature, repectively (K), an c i the intrument enitivity (V / Wm -2 ). Uing thee aumption, the temperature of the enor an the net flux arriving at the enor urface are erive from meaurable quantitie; that i T = Tb (2.9) an Fnet =. (2.10) c Equation 2.7 i olve for the quantity of interet, E, which reult in Equation 2.7a. The unknown quantitie, F net an T, are replace with meaurable quantitie, T b an U emf in Equation 2.7b. The lat two term of Equation 2.7b are then expane. The quantity T b i on the orer of 300 K an U emf / i typically le than 3 K. The thir term of the expanion i le than 0.06 percent of the firt, o the firt two term of the expanion are ufficient to approximate the expanion. Uner thi approximation, Equation 2.7 become E Fnet σt ρσt σt =, (2.7a) α U U emf emf σt ρσ ( Tb ) σ ( Tb ) or E =, (2.7b) cα α 1 σ 1 αρ 3 σ 1 αρ σ or finally, E = Tb Tb T cα α. (2.7c) α α ( ) ( ) With the incluion of the greyboy aumption (α = ), Equation 2.7c become Equation Moel Implication The analytical formulation of the thermal exchange within the pyranometer offer inight into the information neee to correct meaurement. Equation 2.7c ugget that an

7 Amie M. mith Chapter 2. Analytical Moel of Thermal Exchange in the Pyranometer 17 intrument may be calibrate to account for thermal effect when the inner ome temperature an the intrument boy temperature are monitore. Becaue of it large thermal ma, the intrument boy temperature i aume to be uniform throughout, an changing only lowly. Therefore, it temperature meaurement location i not critical, but it houl be taken a cloe to the col junction of the thermopile a poible to avoi incluing any graient that may be preent in the boy. Monitoring the ome temperature oe poe problem. Firt, the ome oe not have a ingle uniform temperature. Rather, a temperature itribution i preent that varie from the intrument boy temperature at the bae of the gla to an extreme temperature at the vertex of the ome, or near the vertex if the ome i expoe to a non-uniform environment. It i hypotheize that a ingle temperature can repreent the itribution in Equation 2.7c. Then the net irraiance arriving at the urface of the enor from a thermally uniform ome at thi temperature i the ame a that arriving from the ome with a graient of temperature. A location for meauring thi temperature i etermine by examining temperature itribution that exit uner variou conition, an then ientifying a repreentative temperature on the itribution. Another concern in meauring the ome temperature i a tranient effect. The ome have a mall thermal ma, an their temperature may vary or fluctuate quickly. The poibility exit that the thermopile meauring a temperature may not recor the thermal hitory with much preciion becaue the thermopile itelf ha thermal ma an it lea may conuct heat to or from the attachment point to the ome. A finite element moel of the pyranometer i helpful in etermining an appropriate location for the ome temperature meaurement. Repreenting the phyical ytem numerically will provie a etaile ecription of the ome temperature graient, which may be interprete a ecribe above to etermine an ieal location to monitor the repreentative ome temperature. Chapter 3 ecribe the contruction of a numerical moel of the pyranometer ome. Thi moel i ue to etermine the ieal location to meaure the repreentative temperature uner variou conition.