Thermophysical and transport properties of humid air at temperature range between 0 and 100 C

Transcription

1 Avalable onlne at Energy Converson and Management 49 (008) Thermophyscal and transport propertes of humd ar at temperature range between 0 and 100 C P.T. Tslngrs * Department of Energy Engneerng, Technologcal Educaton Insttuton (TEI) of Athens, A. Spyrdonos Street, GR 1 10 Egaleo, Athens, Greece Receved 4 February 007; accepted 9 September 007 Avalable onlne 8 November 007 Abstract The am of the present nvestgaton s evaluaton of the thermophyscal and transport propertes of most ar as a functon of mxture temperature wth relatve humdty as a parameter, rangng between dry ar and saturaton condtons. Based on a lterature revew of the most wdely avalable analytcal procedures and methods, a number of developed correlatons are presented, whch are employed wth recent gas mxture component propertes as nput parameters, to derve the temperature and humdty dependence of mxture densty, vscosty, specfc heat capacty, thermal conductvty, thermal dffusvty and Prandtl number under condtons correspondng to the total barometrc pressure of kpa. The derved results at an accuracy level sutable for engneerng calculatons were plotted and compared wth adequate accuracy wth exstng results from prevous analytcal calculatons and measured data from earler expermental nvestgatons. The saturated mxture propertes were also approprately ftted, and the fttng expressons sutable for computer calculatons are also presented. Ó 007 Elsever Ltd. All rghts reserved. Keywords: Thermophyscal propertes; Densty; Vscosty; Specfc heat capacty; Thermal conductvty; Thermal dffusvty; Prandtl number 1. Introducton Although a consderable amount of effort has been devoted durng the last several decades towards evaluaton of the thermophyscal and transport propertes of dry ar and water vapor for a wde range of temperatures, a relatvely lmted attenton was orented toward nvestgaton of the correspondng propertes of humd ar. The development of methods for evaluaton of humd ar propertes was the subject of a number of earler nvestgatons, whch were employed to conduct property evaluaton calculatons at specfc temperature regons of nterest n a certan range of scentfc and technologcal applcatons, lke metrology and calbraton as well as for ar condtonng. These scentfc felds of applcaton and the correspondng nvestgatons manly refer ether to low temperatures lke those carred out by Gacomo [1], Daves [], Zuckerwar and * Tel.: E-mal address: ptslng@teath.gr Meredth [3], Rasmussen [4], Hyland and Wexler [5] or to relatvely hgher temperatures as those by Mellng et al. [6], who nvestgated most ar propertes n the temperature range between 100 and 00 C. However, the knowledge of thermophyscal and transport propertes at ntermedate temperature levels up to 100 C s vtal for certan other technologcal felds, lke dryng and water dstllaton, to allow accurate predcton of heat and mass transfer phenomena durng the physcal processes nvolved. Wth the excepton of a bref report from a survey on materal propertes n SI unts by Nelson [7], nvestgatons on the development of correlatons for dervaton of the transport propertes of humd ar for such calculatons are lackng from the lterature. When approprate most ar data are not readly avalable, t s common practce to nvolve dry, nstead of most ar propertes n the assocated heat transfer calculatons. However, the dry ar assumpton cannot always be tolerated, snce the use of dry ar nstead of most ar propertes may possbly lead to consderable errors n predctng heat /$ - see front matter Ó 007 Elsever Ltd. All rghts reserved. do: /j.enconman

2 P.T. Tslngrs / Energy Converson and Management 49 (008) Nomenclature a, b, c numercal constants A, B vral coeffcents A 0, A 1, A, A 3 numercal constants B 0, B 1, B, B 3 numercal constants c p specfc heat capacty (J/kg K) C 1, C, C 3 numercal constants CA 0,CA 1,...,CA 4 numercal constants CV 0,CV 1,CV numercal constants COD coeffcent of determnaton E 0, E 1, E, E 3, E 4 numercal constants f enhancement factor k thermal conductvty (W/m K) K 1, K, K 3 numercal constants KA 0,KA 1,...,KA 5 numercal constants KV 0,KV 1,KV numercal constants M molar mass (kg/kmol) MA 0,MA 1,...,MA 4 numercal constants MV 0,MV 1 numercal constants n mole number P pressure (Pa) Pr Prandtl number R unversal gas constant ( J/mol K) RH relatve humdty SD 0,SD 1,...,SD 3 numercal constants SV 0,SV 1,...,SV 4 numercal constants SC 0,SC 1,...,SC 6 numercal constants SA 0,SA 1,...,SA 5 numercal constants SK 0,SK 1,...,SK 5 numercal constants S,SP 1,...,SP 4 numercal constants t temperature ( C) T absolute temperature (K) z compressblty factor Greek letters a thermal dffusvty (m /s) D dfference e dmensonless multpler H nteracton parameter l vscosty (Ns/m ) n 1, n 1 dmensonless multplers q densty (kg/m 3 ) U nteracton parameter Subscrpts a ar deal m mxture sv saturated vapor tr monoatomc value v vapor 0 total and mass transport fluxes owng to the relatvely low molar mass of water vapor. Ths s partcularly vald, especally at elevated temperatures and close to saturaton condtons, when substantal amounts of water vapor are present n the mxture. A comprehensve revew of the avalable lterature has shown an almost total lack of bblographcal reports referrng to a complete account of most ar thermophyscal propertes as a functon of relatve humdty under atmospherc pressure condtons for the temperature range of nterest. Although a relatvely lmted number of theoretcal analyses and developed correlatons for humd ar transport propertes sporadcally appeared n the lterature, the avalable data from expermental measurements are very scarce and ncomplete, referrng to certan fxed specfc temperature levels, based almost entrely on earler nvestgatons. The am of the present nvestgaton s to derve a full account of the commonly employed thermophyscal propertes of humd ar at the temperature range between 0 and 100 C as a functon of the mxture degree of saturaton and temperature for ordnary heat transfer engneerng calculatons. For ths purpose, the present analyss was based on the correspondng thermophyscal propertes of dry ar and water vapor from contemporary well establshed lterature sources, as well as on the wdely acceptable theoretcal procedures n the exstng lterature. Along wth the dervaton of approprate analytcal formulatons, the results correspondng to saturated mxture condtons are also presented n the form of approprately developed polynomal fttng expressons sutable for computerzed calculatons. The derved results from the present nvestgaton were also compared wth correspondng data derved from exstng mathematcal correlatons and wth measured data from earler nvestgatons.. The evaluaton of thermophyscal and transport propertes of humd ar For the purpose of determnng ts thermophyscal and transport propertes, humd ar s regarded as a bnary mxture of dry ar and water vapor. The molar fracton of water vapor s defned as the rato of water vapor moles to the total number of moles of the mxture as, x v ¼ n v ¼ n v ¼ P v ð1þ n m n a þ n v The relatve humdty s defned as, RH ¼ n v ¼ x v ¼ P v n sv x sv from whch, ðþ

3 1100 P.T. Tslngrs / Energy Converson and Management 49 (008) x v ¼ x sv RH ð3þ where x sv s the molar fracton of the saturated water vapor under the assumpton of deal gas behavor. However, owng to nteracton effects between real gas molecules, a small ncrease of the saturaton vapor pressure n ar s developed, whch s taken nto account by the ntroducton of an approprate correctve numercal factor, known as an enhancement factor [1,8]. Ths, beng a functon of pressure and temperature, s ntroduced to correct the molar fracton of saturated vapor pressure, x sv ¼ f ðp; T Þ ð4þ The molar fracton of water vapor s then calculated from Eqs. (3) and (4) as a functon of the total atmospherc pressure and the saturated vapor pressure at a specfc temperature by the followng expresson, x v ¼ f ðp; T ÞRH ð5þ The evaluaton of the numercal value of the enhancement factor was the subject of varous earler nvestgatons, lke those by Hyland and Wexler [9] and Hyland [10]. However, for the purpose of the present analyss, ts calculaton was performed accordng to Hardy [11] by the followng smplfed fttng expresson recommended by Greenspan [1] f ðp; T Þ¼exp n 1 1 wth n 1 ¼ X3 ¼0 þ n 1 ð6þ A T ð7þ " # n ¼ exp X3 T ¼0 B ð8þ The numercal values of the constants n Eqs. (7) and (8) correspondng to the temperature range between 0 and 100 C are A 0 = , A 1 = , A = , A 3 = , B 0 = , B 1 = , B = and B 3 = Typcal calculated values of the enhancement factor as a functon of temperature, whch are very close to unty, are shown n Table 1. Although these numercal values appear to be lower than those derved from Gacomo [1] correspondng to an apprecably lower temperature range between 0 and 30 C, they are very slghtly hgher than unty, so the unty value assumpton of the enhancement factor f = 1 leads to less than 0.5% maxmum error at temperatures around 75 C, whch appears to be qute acceptable for the purpose of ordnary calculatons. Although ths assumpton for the condtons correspondng to the present nvestgaton leads to nsgnfcant errors for ordnary engneerng calculatons, for any other condtons, ts valdty should always be properly justfed. Extensve nvestgatons have been also conducted durng the last several decades on the saturaton vapor pressure of water, as reported by Alduchov and Eskrdge [13], manly from the clmatologcal and atmospherc research communtes. However, these are manly restrcted to a temperature range between sub-freezng up to 40 C, whch s rrelevant to the temperature range of nterest. It was, therefore, decded, for the purpose of the present nvestgaton, to derve values of by the followng fourth degree polynomal, whch was developed by fttng the saturaton vapor pressure data between 0 and 100 C drectly from the thermodynamc propertes of water [14] accordng to the followng expresson, ¼ E 0 þ E 1 t þ E t þ E 3 t 3 þ E 4 t 4 ð9þ where s n kpa, for the followng values of numercal constants E 0 = , E 1 = , E = , E 3 = and E 4 = The prevous polynomal ft expresson typcally offers better than 1.5% accuracy for 5 C, whch mproves to about 0.% for temperatures up to 100 C..1. Densty The evaluaton of most ar densty was the subject of several prevous nvestgatons coverng varous scentfc felds of applcatons (1, and 5, 6). The densty of the bnary mxture of pure water vapor and dry ar at the correspondng partal pressures and molar fractons of P v, x v and P a = P v, x a =1 x v, respectvely, s calculated wth suffcent accuracy through the gas equaton of state, by the followng smple mxng correlaton, 1 q m ¼ z m ðx v ; T Þ R T M a P 0 P v P v þ M v ð10þ where z m (x v, T) s the compressblty factor for the gas mxture. From the above expresson, the densty can be derved as a functon of the molar fracton of water vapor as 1 q m ¼ z m ðx v ;T Þ R T M a 1 x v 1 M v ð11þ M a whch, combned wth Eq. (5), leads to the followng expresson for the densty of the bnary mxture, Table 1 The calculated enhancement and compressblty factors for temperatures up to 100 C T ( C) f z v

4 P.T. Tslngrs / Energy Converson and Management 49 (008) q m ¼ z m ðx v ; T Þ R T M a 1 f ðp; T ÞRH 1 M v M a ð1þ Although the compressblty factor for dry ar at ambent condtons s unty, Mellng et al. [6] derved ts value for water vapor as a root mean square ft of data avalable from the lterature and calculated the compressblty factor of humd ar between 100 and 00 C by the followng approxmate mxng expresson, a þ c T z m ðx v ; T Þ¼1 þ x v 1 þ b T 1 ð13þ wth the numercal values of the constants a, b and c approprately selected for the specfed temperature range of nterest. Although the calculated values of z m from Eq. (13) were found to be close to unty, rangng between and , the compressblty factor for the purpose of the present nvestgaton, referrng to the temperature regon between 0 and 100 C, was evaluated from the vral equaton of state accordng to the followng expresson, z v ¼ 1 þ A þ B P sv ð14þ whch was recommended by Hyland and Wexler [8]. The second and thrd pressure seres vral coeffcents were calculated by, A ¼ C 1 þ C e C 3 T B ¼ K 1 þ K e K 3 T ð15þ ð16þ wth C 1 = Pa 1, C = Pa 1, C 3 = (K 1 ), K 1 = Pa, K = Pa and K 3 = K 1. Typcal calculated values of z v, whch were found to be very close to unty, are shown n Table 1 as a functon of temperature. Fxng the compressblty factor to the unty value z v =1 leads to very small errors for the calculaton of mxture densty, typcally about 0.38% for 50 C and less than about 1.5% for 100 C. The combned effect of fxng both the enhancement and compressblty factors at the unty value s estmated to be responsble for an overall maxmum error for the evaluaton of densty that ranges between about 0.4% and less than 1.5% at the correspondng temperatures of 0 and 100 C. Although the assumpton of a unty value for the compressblty and enhancement factors leads to nsgnfcant errors for the purpose of the present analyss, ts valdty should always be properly justfed for specfc condtons, when sacrfce of mproved accuracy cannot be tolerated... Vscosty Based on further knetc theory approach consderatons, Red et al. [15] recommended the followng expresson for the vscosty of a mxture of dlute gases wth components, whch was based on earler nvestgatons by Wlke [16], l m ¼ Xn ¼1 x l Pn j¼1 x ju j wth the nteracton parameters U j and U j, gven by, U j ¼ U j ¼ l j l ð17þ 1= 1=4 1 þ l M l j j M h 1= ð18þ 8 1þ M M j M M j U j ð19þ As derved from Eq. (18), U = U jj = 1, whch, combned wth Eq. (17), after ts expanson, leads to the followng expresson, ð1 x v Þl l m ¼ a x v l þ v ð1 x v Þþx v U av x v þð1 x v ÞU va ð0þ Takng nto consderaton Eq. (5), the prevous expresson becomes, h P 1 f ðp;t ÞRH sv l a l m ¼ h P 1 f ðp;t ÞRH sv þ f ðp;t ÞRH P f ðp;t ÞRH sv l v þ h f ðp;t ÞRH þ 1 f ðp;t ÞRH l v U av :U va ð1þ whch offers the vscosty of a humd ar mxture at a specfc temperature and relatve humdty as a functon of dry ar and water vapor vscostes for the followng values of the nteracton parameters, pffff U av ¼ 4 1 þ M " 1 a 1 þ l 1 # 1 a M 4 v M v M ðþ a pffff U va ¼ 4 1 þ M " 1 v 1 þ l 1 # 1 v M 4 a M a M ð3þ v.3. Thermal conductvty Red et al. [15] suggests the followng expresson, whch was orgnally proposed by Wassljewa [17], as the bass of the calculaton of the thermal conductvty of the mxture, k m ¼ Xn x k P n ¼1 j¼1 x ð4þ jh j Based also on the orgnal nvestgatons by Mason and Saxena [18], they have recommended that, l a

5 110 P.T. Tslngrs / Energy Converson and Management 49 (008) H j ¼ e 1=4 1= 1 þ ktr k trj M M j h 1= ð5þ 8 1þ M M j where the rato of the monoatomc values of thermal conductvty n the prevous expresson s calculated accordng to Ref. [15] by, k tr k trj ¼ l l j M j M ð6þ Mason and Saxena [18] suggested that although e can be a complex functon of several knetc parameters, ts numercal value, whch s never far from unty, can be set equal to some best average value, whch they recommended to be e = for non-polar gases. Tondon and Saxena [19], suggested that better accuracy can be obtaned by fxng ths numercal value to e = 0.85 for a mxture of polar and non-polar gases, whle accordng to Red et al. [15], ths numercal constant was recommended to be fxed at ts unty numercal value e = 1, whch was also adopted here for the purpose of the subsequent calculatons. Under ths assumpton, the substtuton of Eq. (6) nto Eq. (5) leads to, U j ¼ H j ð7þ The followng expresson, employed for calculaton of the thermal conductvty of mxtures, s derved from Eq. (4), ð1 x v Þk a x v k v k m ¼ þ ð8þ ð1 x v Þþx v U av x v þð1 x v ÞU va Takng nto consderaton Eq. (5), the prevous expresson becomes, h P 1 f ðp;t ÞRH sv k a k m ¼ h P 1 f ðp;t ÞRH sv þf ðp;t ÞRH P f ðp;t ÞRH sv k v þ h f ðp;t ÞRH þ 1 f ðp;t ÞRH U av U va ð9þ Ths expresson s very smlar to the correspondng Eq. (1) for calculaton of the mxture vscosty, and t s employed for evaluaton of the mxture thermal conductvty as a functon of the correspondng values of the humd ar components..4. Specfc heat capacty The general procedure for evaluatng the specfc heat capacty s to apply a smple lnear mxng equaton as proposed by Wong and Embelton [0], who derved the heat capacty of humd ar as a functon of relatve humdty for the lmted temperature range between 0 and 30 C. The same procedure was also adopted by several other nvestgators, lke Zuckerwar and Meredth [3], Durst et al. [1], although for a non-clearly specfed temperature range, and Rasmussen [4], who derved correlatons for a relatvely narrow temperature range around ambent temperatures, as well as Mellng et al. [6], who reported data between 100 and 00 C. Followng the same approach, the specfc heat capacty of the deal gas mxture can be expressed as M a M v c pm ¼ c pa x a þ c pv x v ð30þ M m M m To account for the real gas behavor, the correcton term Dc p was proposed accordng to Red et al. [15], c p;m c p;m ¼ Dc p ð31þ whch s a complex functon of the frst- and second-order devaton functons and the acentrc factor of the mxture molecules. Ths factor represents the nfluence of acenrtcty or non-sphercty of the consttuent gas molecules, whch, although for monoatomc gases t s zero, ncreases wth molecular weght and molecular structure complexty wth respect to both geometry and polarty. The numercal value of the correcton term Dc p was found, accordng to Mellng et al. [6] for the specfc condtons, to be comparatvely small, leadng to the maxmum 1.5% correctons to deal gas behavor, and therefore, ts effect was assumed to be neglgble, leadng to c pm = c pm for the purpose of the subsequent calculatons. However, takng nto account Eq. (5), the molar fracton of dry ar s x a ¼ 1 x v ¼ 1 f ðp; T ÞRH and snce ð3þ M m ¼ M a x a þ M v x v ð33þ Eq. (30) becomes, h c pa 1 f ðp;t ÞRH M a þc pv f ðp;t ÞRH M v c pm ¼ h M a 1 f ðp;t ÞRH þm v f ðp;t ÞRH.5. Thermal dffusvty ð34þ Thermal dffusvty s calculated from ts defnton expresson, k m a m ¼ ð35þ q m c pm takng nto account the prevously derved Eqs. (1), (9) and (34) for the respectve calculatons of densty, thermal conductvty and specfc heat capacty of humd ar..6. Prandtl number Ths dmensonless number s defned as Pr m = m m /a m. Snce the thermal dffusvty from Eq. (35) and the kne-

6 P.T. Tslngrs / Energy Converson and Management 49 (008) matc vscosty, as derved from l m = q m Æ m m, are both functons of densty, the Prandtl number s evaluated as a functon of known thermophyscal propertes l m, c pm and k m,as Pr m ¼ l m c p;m ð36þ k m wth the values of propertes l m, c p, m and k m derved from Eqs. (1), (34) and (9), respectvely. 3. The thermophyscal and transport propertes of dry ar and water vapor Dry ar s a mxture of several gas components at dfferent concentratons, wth a composton that can approxmately be consdered to be constant n the atmosphere. As soon as the molar mass M a, and the molar fracton x a, of the n ndvdual consttuent gases of dry ar mxture are known, the molar mass of atmospherc ar can be derved by the followng expresson, P n ¼1 M a ¼ x a; M P a; n ¼1 x ð37þ a; Assumng a composton smlar to that of standard dry ar, as descrbed for example by Ref. [], a molar mass of M a = kg/kmol s derved from the prevous expresson. The calculaton of the thermodynamc and transport propertes of dry ar and water vapor mxtures s based on the exstng propertes of the consttuent gases, both of whch have been the objectve of extensve research durng the last several decades. Snce the selecton of these propertes as nput data for the subsequent calculatons s crucal for accurate evaluaton of the correspondng humd ar propertes, one has to rely on recent, accurate and relable data from one among the several avalable, well establshed lterature sources. From ths pont of vew, all selected propertes were derved from the Handbook of Heat Transfer [3], n whch data for the vscosty, thermal conductvty and specfc heat capacty of dry ar are avalable, as compled from Irvne and Lley [4]. The vscosty of dry ar n Ns/m 10 6 s offered by the followng correlaton, l a ¼ MA 0 þ MA 1 T þ MA T þ MA 3 T 3 þ MA 4 T 4 ð38þ n the temperature range 3 C 6 t 6 37 C, for the followng values of the numercal constants, MA 0 = , MA 1 = , MA = , MA 3 = and MA 4 = The thermal conductvty of dry ar n W/m K 10 3 s expressed by the followng correlaton, k a ¼ KA 0 þ KA 1 T þ KA T þ KA 3 T 3 þ KA 4 T 4 þ KA 5 T 5 ð39þ for KA 0 = , KA 1 = , KA = , KA 3 = , KA 4 = and KA 5 = Ths expresson s vald at the temperature range between 3 C 6 t C. The specfc heat capacty of dry ar n kj/kg K, whch s vald for the same temperature range, s gven by the followng expresson c pa ¼ CA 0 þ CA 1 T þ CA T þ CA 3 T 3 þ CA 4 T 4 ð40þ for the followng values of the numercal constants CA 0 = , CA 1 = ,CA = , CA 3 = and CA 4 = Correspondng data for the water vapor propertes were taken from the same lterature source as compled from Toulukan et al. [5]. These data were ftted by the followng approprate degree polynomals n the temperature range 0 6 t6 10 C. The vscosty n Ns/m 10 6 was determned by the followng lnear expresson, l v ¼ MV 0 þ MV 1 t ð41þ wth MV 0 = and MV 1 = The thermal conductvty n W/m K 10 3 was determned by the expresson, k v ¼ KV 0 þ KV 1 t þ KV t ð4þ where KV 0 = , KV 1 = and KV = The specfc heat capacty n (kj/kg K) was determned by the followng expresson, c pv ¼ CV 0 þ CV 1 t þ CV t ð43þ wth CV 0 = , CV 1 = and CV = Results and dscusson The derved densty from Eq. (1) s plotted n Fg. 1 as a functon of temperature for an ncreasng relatve humdty from the mnmum value of 0%, correspondng to dry ar (top curve), up to the maxmum value of RH = 100% (lower curve), correspondng to saturated condtons, n 10% steps. The ncrease of relatve humdty leads to a decrease of humd ar densty, especally at the range of hgher temperatures, owng to M a M v > 0. Although a relatve humdty ncrease between 0 and 100% leads to an almost neglgble densty decrease of humd ar at near freezng temperatures, t s responsble for a densty decrease of about 4.8% at 50 C, and to a sgnfcant reducton of humd ar densty of about 37.5%, at temperatures close to 100 C.

7 1104 P.T. Tslngrs / Energy Converson and Management 49 (008) rangng between 0% and 100% n 10% steps, as derved from Eq. (1). Although the ncrease of relatve humdty up to ts saturaton level leads to nsgnfcant decreases of vscosty at temperatures close to 0 C, ts effect leads to approxmately 6.8% decrease at temperatures around 50 C and to a decrease of approxmately 45% at the temperature of 100 C. For each of the constant relatve humdty curves, there s a temperature value correspondng to a maxmum vscosty, whch moves towards hgher temperatures as the relatve humdty decreases. It can be seen that although the temperature for the maxmum humd ar vscosty of 10 5 Ns/m at the mxture relatve humdty of RH = 0% s about 86 C, t sgnfcantly decreases to about 40 C for the maxmum vscosty of Ns/m correspondng to RH = 100%. The saturated mxture vscosty for the temperature range between 0 and 100 C was ftted by the followng fourth degree polynomal, Fg. 1. The most ar densty as a functon of temperature wth the relatve humdty as a parameter rangng between dry ar (top curve RH = 0%) and saturaton condtons (lower curve RH = 100%) n 10% steps. The densty of the saturated mxture for the temperature range between 0 and 100 C was ftted by the followng thrd degree polynomal, q m ¼ SD 0 þ SD 1 t þ SD t þ SD 3 t 3 ð44þ wth a coeffcent of determnaton and the values of the numercal constants SD 0 to SD 3 as shown n Table 4. The humd ar vscosty s plotted n Fg. as a functon of temperature for parametrc values of relatve humdty l m ¼ SV 0 þ SV 1 t þ SV t þ SV 3 t 3 þ SV 4 t 4 ð45þ wth the correspondng coeffcent of determnaton and the values of the numercal constants SV 0 SV 4 shown n Table 4. The thermal conductvty of humd ar as derved from Eq. (9) s plotted n Fg. 3 for the parametrc values of relatve humdty rangng between 0% (top curve) and 100% (lower curve) n fxed 10% relatve humdty steps. A smlar effect of the relatve humdty ncrease between 0% and 100% s demonstrated as before, whch, although at the lower temperatures leads to a neglgble decrease, at temperatures around 50 C leads to a decrease of 3.5% and Fg.. The vscosty of most ar as a functon of temperature wth the relatve humdty as a parameter rangng between dry ar (top curve RH = 0%) and saturaton condtons (lower curve RH = 100%) n 10% steps. Fg. 3. The thermal conductvty of most ar as a functon of temperature wth the relatve humdty as a parameter rangng between dry ar (top curve RH = 0%) and saturaton condtons (lower curve RH = 100%) n 10% steps.

8 k m ¼ SK 0 þ SK 1 t þ SK t þ SK 3 t 3 þ SK 4 t 4 ð46þ at temperatures around 100 C to a sgnfcant decrease of about 1.5% n thermal conductvty. A maxmum value of thermal conductvty s developed for each fxed relatve humdty curve, whch moves towards hgher temperatures as the relatve humdty decreases. Ths maxmum moves typcally from the temperature of 63 C to about 94 C as the relatve humdty decreases from saturaton level condtons to about RH = 40%. The thermal conductvty of the saturated mxture for the temperature range of nterest was ftted by the followng fourth degree polynomal, P.T. Tslngrs / Energy Converson and Management 49 (008) wth a coeffcent of determnaton and the values of the numercal constants SK 0 SK 4 as shown n Table 4. The specfc heat capacty of humd ar as derved from Eq. (34) s plotted n Fg. 4 wth the relatve humdty as a parameter, ncreasng from the value of RH = 0% (lower lne) to 100% (top lne) n 10% steps. Agan, although the ncrease of relatve humdty up to ts saturaton level leads to an almost neglgble ncrease of vscosty at temperatures close to 0 C, ts effect leads to an ncrease of about 7% and 100% at the correspondng temperatures of 50 and 100 C, respectvely. The saturated mxture specfc heat capacty for the temperature range between 0 and 100 C was ftted by the followng ffth degree polynomal, c pm ¼ SC 0 þ SC 1 t þ SC t þ SC 3 t 3 þ SC 4 t 4 þ SC 5 t 5 ð47þ wth the correspondng coeffcent of determnaton and values of the numercal constants SC 0 to SC 5 shown n Table 4. Fg. 5. The thermal dffusvty of most ar as a functon of temperature wth the relatve humdty as a parameter rangng between dry ar (top curve RH = 0%) and saturaton condtons (lower curve RH = 100%) n 10% steps. The thermal dffusvty of humd ar s plotted n Fg. 5 as derved from Eq. (35) for an ncreasng parametrc value of relatve humdty between RH = 0% (top lne) to 100% (lower lne) n unform 10% steps. The correspondng saturated mxture propertes for the temperature range of nterest were ftted by the followng fourth degree polynomal, a m ¼ SA 0 þ SA 1 t þ SA t þ SA 3 t 3 þ SA 4 t 4 ð48þ wth a coeffcent of determnaton and the values of the numercal constants SA 0 to SA 4 as shown n Table 4. Fg. 4. The specfc heat capacty of most ar as a functon of temperature wth the relatve humdty as a parameter rangng between dry ar (top curve RH = 0%) and saturaton condtons (lower curve RH = 100%) n 10% steps. Fg. 6. The Prandtl number of most ar as a functon of temperature wth the relatve humdty as a parameter rangng between dry ar (lower curve RH = 0%) and saturaton condtons (top curve RH = 100%) n 10% steps.

9 1106 P.T. Tslngrs / Energy Converson and Management 49 (008) Fnally, the temperature dependence of the most ar Prandtl number for a relatve humdty rangng between RH = 0% (lower curve) to 100% (top curve) as a parameter s shown n Fg. 6. A smlar behavor of the growng dependence of both propertes, the thermal dffusvty and Prandtl number wth the relatve humdty, especally at elevated mxture temperatures s exhbted as shown n Fgs. 5 and 6. The Prandtl number for the saturated mxture for the same temperature range was ftted by the followng fourth degree polynomal, Pr m ¼ S þ SP 1 t þ SP t þ SP 3 t 3 þ SP 4 t 4 ð49þ wth a coeffcent of determnaton and correspondng numercal constants as shown n Table Comparsons wth the results from earler nvestgatons Nelsons report [7] wth the proposed emprcal correlatons appears to be the sngle readly avalable source of formulae sutable for the dervaton of most ar thermophyscal propertes to ft data for the temperature range of nterest. However, references or ctatons on the orgn of the recommended expressons, except for Ref. [6], are mssng and detaled dervatons are completely lackng from hs orgnal paper. The recommended expresson for mxture densty s, q m ¼ð3:484 1:317 x v Þ ð50þ 73:15 þ t As derved from a careful nspecton, the prevous expresson can easly be derved drectly from Eq. (1), takng nto consderaton Eq. (5) and assumng unty values for z m and f(p, T), as well as the standard numercal values of the followng constants R = J/mol K, M a = kg/kmol and M v = 18.0 kg/kmol, leadng n ths way to dentcal results wth those of Eq. (1). The proposed expresson for mxture vscosty s, l m ¼ l a þ x v ð0:7887 l v l a Þ ð51þ 1 0:113 x v for whch the report referred to Ref. [6], where ts dervaton s attrbuted to an earler cted work by Hernng and Zpperer recommendng the dervaton of mxture vscosty as, P n ¼1 l m ¼ x l ðm Þ 1= P n ¼1 x ð5þ ðm Þ 1= The above expresson, appled to a two component system, namely dry ar and water vapor, s dentcal to Eq. (9) n Ref. [6], as proposed by Krscher and Kast [7], for the same molecular weghts of mxture components. Ths expresson s completely dfferent from Eq. (1), whch was employed for the purpose of the present nvestgaton. The proposed expresson for the specfc heat capacty, although of an unspecfed orgn s, c pm ¼ c pa þ 0:6 P v P v ð53þ It can easly be seen that ths expresson can be derved from Eq. (34) based on lnear mxng consderatons by omttng the water vapor pressure contrbuton quantty M v Æ f(p, T) Æ RH Æ ( / ) of the denomnator and assumng a unty value for the water vapor specfc heat capacty, whch, however, for the temperature regon of nterest, ranges between about kj/kg K. Owng to the fact that these assumptons are not vald, Eq. (53) appears to be of questonable accuracy. The proposed thermal conductvty expresson n (mw/ m K) unts s, k m ¼ k a þ x v ð0:8536 k v k a Þ ð54þ 1 0:1464 x v Ths s also attrbuted to the chemcal engneers handbook [6] n whch ts dervaton s attrbuted to the followng expresson, recommended by Frend and Adler [8], P n ¼1 k m ¼ x k ðm Þ 1=3 P n ¼1 x ðm Þ 1=3 ¼ x a k a ðm a Þ 1=3 þ x v k v ðm v Þ 1=3 ð55þ x a ðm a Þ 1=3 þ x v ðm v Þ 1=3 from whch Eq. (54) s derved for the numercal constants of M a = kg/kmol and M v = 18.0 kg/kmol, whch s completely dfferent than the derved Eq. (9). The exstng expermental measurements of mxture thermophyscal propertes n the lterature are scarce, based mostly on earler nvestgatons conducted at certan specfc temperature levels for a relatvely wde range of water vapor molar fracton. Snce the current nvestgaton covers mxture propertes at a constant pressure correspondng to normal atmospherc condtons, the comparatve presentaton of the derved results and exstng measured data requres the reference of the water vapor molar fracton of the measurements to the correspondng relatve humdty of the mxture up to ts saturaton level. Ths was calculated from the correspondng molar fracton and the saturated vapor pressure at the specfed temperature based on Eq. (5) for a unty value of the enhancement factor. The exstng vscosty measurements, as compled from the lterature, are shown n Table wth the correspondng approxmate evaluated relatve humdty levels rounded to ther nearest nteger value. The measured data from Kestn and Whtelaw, Hochraner and Munczak and Vargaftk [9] were taken from the Tables 5 to 7 of Ref. [6], whle results derved from Mason and Monchck were compled graphcally from plotted data by Kestn and Whtelaw [30]. The thermal conductvty data for the temperature level of 80 C were evaluated from measured (k m /k a ) ratos orgnally derved by Gruss and Schmck [31] and from publshed data by Vargaftk [9], whle for 60 C, they were taken from Touloukan et al. [3] as reported n Ref. [6].

10 P.T. Tslngrs / Energy Converson and Management 49 (008) Table Data correspondng to vscosty measurements from varous lterature sources ( 10 6 kg/m s) From Kestn and Whtelaw [6] 5 C % 5.5% 35% 51% 54% 50 C % 0% 5% 34% 51% 98% 75 C % 0% 5% 35% 51% 70% 83% 97% 100% From Hochraner and Munczak [6] 0 C % 6% 8% 93% 30 C % 41% 61% 81% 9% 40 C % 0% 41% 63% 83% 93% 50 C % 1% 41% 61% 8% 93% From Vargaftk [6,9] 50 C % 8% 60 C % 51% 100% 70 C % 3% 65% 97% 80 C % 1% 43% 64% 86% 100% 90 C % 14% 9% 43% 58% 7% 87% 100% 100 C % 10% 0% 30% 40% 50% 60% 70% 80% 90% 100% From Mason and Monchck [30] 5 C % 100% 50 C % 8% 100% 75 C % 6% 53% 79% 100% In Fg. 7, the derved vscosty, represented by thck sold lnes of ncreasng relatve humdty n unform 10% steps, s compared to the correspondng results, plotted wth broken lnes, calculated through Eq. (43) as proposed by Nelson and to varous measurements from the lterature. It appears that although not dentcal, there s a remarkable agreement between the results from the present nvestgaton and Nelson s emprcal correlaton, the maxmum dfference beng typcally less than about 4% and manly confned to dfferent slopes between unform relatve humdty curves. The dscrete data ponts, representng earler measurements for certan correspondng fxed temperatures, cover almost the entre relatve humdty range, approxmately up to the saturaton level, n ncreasng although arbtrary steps, as can also be seen from Table. The measured data appear to be slghtly hgher than both the calculated mxture vscosty from the present nvestgaton and the results from Nelson s correlaton, especally at the md temperature range of nterest, the devatons beng typcally lower than 8%. The correspondng results referrng to thermal conductvty are also shown n Fg. 8 n whch the devatons between the results from the present nvestgaton and Nelsons data agan appear to be small, typcally about 1%, and mostly confned to slght slope dfferences between correspondng fxed relatve humdty curves over the entre temperature range of nterest. Thermal conductvty measurements n the lterature appear to be very scarce,

11 1108 P.T. Tslngrs / Energy Converson and Management 49 (008) Fg. 7. Comparatve vscosty of most ar at varous temperature levels accordng to results from the present analyss (curves n sold lnes), Nelsons correlaton (curves n broken lnes) and dscrete data from earler measurements n the lterature. whle most dfferent reports, lke Vargaftk [9], Touloukan [3] and Tondon and Saxena [19], repeatedly cte the earler measurements by Gruss and Schmck [31], who measured the (k m /k a ) rato at 80 C, from whch, for the purpose of the present nvestgaton, k m was calculated usng dry ar propertes from Ref. [4]. Touloukan [3] also reported measurements from ntrogen/water vapor mxtures, snce ntrogen s the major consttuent of dry ar. The avalable lmted measurements coverng the entre relatve humdty range approxmately up to the saturaton level n arbtrary ncreasng steps, as can be seen n Table 3, were ncluded as dscrete data ponts n Fg. 7. Although these early measurements are nadequate for establshng a complete relable set of valdaton data, they are ndcatng values hgher than those predcted from the present analyss and Nelsons correlaton, by a maxmum level of about 7% correspondng to 65 C and by 10% to 80 C, whch stll compares suffcently well wth theory. Comparatve results for the specfc heat capacty of mxtures as derved from the present analyss (group of thck sold lnes) and accordng to the correlaton Eq. (45) (group of broken lnes) can be seen n Fg. 9, for relatve humdty varyng between 0%, (lower lne n each group) and saturaton condtons (upper lne). Apparently, Nelson s correlaton, owng to the reasons dscussed prevously n detal, leads to unrealstcally hgher values of most ar specfc heat capacty, especally at the hgher relatve humdty and temperature levels. 6. Conclusons Fg. 8. Comparatve thermal conductvty of most ar at varous temperature levels accordng to results from the present analyss (curves n sold lnes), Nelsons correlaton (curves n broken lnes) and dscrete data from earler measurements n the lterature. The present nvestgaton has allowed the dervaton of a complete account of the thermophyscal and transport propertes of most ar n the temperature range between 0 and 100 C under condtons correspondng to normal barometrc pressure of kpa, whch s of major mportance n several felds of technology. The evaluaton of propertes was based on a comprehensve lterature revew for the approprate selecton of the most wdely acceptable procedures and methods as well as dry ar and water vapor thermophyscal propertes, whch were employed as nput parameters for the subsequent calculatons. The methodology and the derved correspondng analytcal correlatons were presented and employed for calculaton of the dry ar and water vapor mxture densty, vscosty, thermal Table 3 Data correspondng to thermal conductvty measurements from varous lterature sources ( 10 W/m K) From Gruss and Schmck [31] 80 C % 15% 3% 37% 4% 48% 53% 65% 67% 95% From Vargaftk [9] 80 C % 1% 43% 65% 86% From Touloukan et al. [3,6] 60 C % 40% 81% >100%

12 P.T. Tslngrs / Energy Converson and Management 49 (008) Table 4 The numercal constants and coeffcents of determnaton (COD) for the proposed polynomal ft expressons for the followng saturated mxture propertes, densty, vscosty, thermal conductvty, specfc heat capacty, thermal dffusvty and Prandtl number Densty (kg/m 3 ) Vscosty (Ns/m ) Thermal conductvty Specfc heat capacty Thermal dffusvty Prandtl number (W/m K) (kj/kg K) (m /s) COD = COD = COD = COD = COD = COD = SD 0 = SV 0 = E-5 SK 0 = E- SC 0 = SA 0 = E-5 S = SD 1 = E- 3 SV 1 = E-8 SK 1 = E-5 SC 1 = E-3 SA 1 = E-7 SP 1 = E- 4 SD = E-5 SV = E- SK = E- SC = E- SA = E-10 SP = E SD 3 = E-7 SV 3 = E-1 SK 3 = E-9 SC 3 = E-6 SA 3 = E- 1 SP 3 = E- 7 SV 4 = E- 14 SK 4 = E- 11 SC 4 = E- 8 SC 5 = E-10 SA 4 = E- 14 SP 4 = E-9 conductvty, specfc heat capacty, thermal dffusvty and Prandtl number data, whch are necessary for performng heat transfer calculatons correspondng to an approprate accuracy level, sutable for ordnary engneerng calculatons. The derved results were graphcally presented for the temperature range of nterest wth the relatve humdty varyng between dry and saturated condtons as a parameter. The saturated mxture propertes were also ftted and the derved fttng expressons sutable for computerzed calculatons were also presented. The results from the present nvestgaton were also compared wth correspondng prevously reported data, ether analytcal or measured, from the lterature. The comparatve presentaton shows that, wth the excepton of the specfc heat capacty, there s very good agreement between the results from the present nvestgaton and prevous analyses and good agreement wth the scarce and mostly ncomplete sporadcally appearng results from earler measurements n the lterature. References Fg. 9. Comparatve specfc heat capacty of most ar at varous temperature levels accordng to results from the present analyss (curves n sold lnes) and Nelsons correlaton (53) (curves n broken lnes) for a relatve humdry rangng between RH = 0% (lower curves) to RH = 100% (top curves). [1] Gacomo P. Equaton for the determnaton of densty of most ar (1981). Metrologa 198;18: [] Daves RS. Equaton for the determnaton of densty of most ar (1981/1991). Metrologa 199;9: [3] Zuckerwar AJ, Meredth RW. Low-frequency absorpton of sound n ar. J Acoust Soc Am 1985;78(3). [4] Rasmussen K. Calculaton methods for the physcal propertes of ar used n the calbraton of mcrophones. Report PL-11b. Department of Acoustc Technology, Techn. Unversty of Denmark; [5] Hyland RW, Wexler A. Formulatons for the thermodynamc propertes of dry ar from K to K and of saturated most ar from to K at pressures to 5 MPa. Trans ASHRAE 1983;89(IIa): [6] Mellng A, Noppenberger S, Stll M, Venzke H. Interpolaton correlatons for flud propertes of humd ar n the temperature range 100 C to 00 C. J Phys Chem Ref Data 1997;6(4): [7] Nelson R. Materal propertes n SI unts, part 4. Chem Eng Prog 1980;76:83. [8] Hyland RW, Wexler A. Formulatons for the thermodynamc propertes of the saturated phases of H O from K to K. Trans ASHRAE 1983;89(IIa): [9] Hyland RW, Wexler A. The second nteracton (cross) vral coeffcent for most ar. J Res NBS 1973;77A(1): [10] Hyland RW. A correllaton for the second nteracton vral coeffcents and enhancement factors for CO -free most ar from 50 to 90 C. J Res Natonal Bureau Standards A Phys Chem 1975;79A(4): [11] Hardy B. ITS-90 formulatons for vapor pressure, frost pont temperature, dew pont temperature and enhancement factors n the range 100 to 100 C. Proc 3rd Int Sympos Humdty Mosture. London: Natonal Physcal Lab (N.P.L.); p [1] Greenspan L. Functonal equatons for the enhancement factors for CO free most ar. J Res Natonal Bureau Standards A Phys Chem 1976;80A(1):41 4. [13] Alduchov OA, Eskrdge RE. Improved magnus form approxmaton of saturaton vapor pressure. Techncal report number DOE/ER/ T6/Nov. USDOE Offce of Energy Research: Washngton, DC; [14] Van Wylen GJ, Sonntag RE. Fundamentals of classcal thermodynamcs. NY: Wley; 1973.

13 1110 P.T. Tslngrs / Energy Converson and Management 49 (008) [15] Red RC, Prausntz JM, Polng BE. The propertes of gases and lquds, Chemcal Engneerng Seres. McGraw Hll Int. Edtons; [16] Wlke CR. A vscosty equaton for gas mxtures. J Chem Phys 1950;18:517. [17] Wassljewa A. Physk Zetshrft 1904;5:734. [18] Mason EA, Saxena SC. Approxmate formula for the thermal conductvty of gas mxtures. Phys Fluds 1958;1(5): [19] Tondon PK, Saxena SC. Calculaton of thermal conductvty of polar nonpolar gas mxtures. Appl Sc Res 1968;19: [0] Wong GSK, Embleton TFW. Varaton of specfc heats and of specfc heat rato n ar wth humdty. J Acoust Soc Am 1984;76(): [1] Durst F, Noppenberger S, Stll M, Venzke H. Influence of humdty on hot wre measurements. Meas Sc Technol 1996;7: [] U.S. Standard Atmosphere U.S. Government Prntng Offce: Washngton, DC; p [3] Rohsenow WM, Hartnett JP, Cho YI. Handbook of heat transfer. 3rd ed. McGraw-Hll; [4] Irvne TF, Lley P. Steam and gas tables wth computer equatons. San Dego: Academc Press; [5] Touloukan YS, Powell RW, Ho CY, Clemend PG. Thermophyscal propertes of matter, vol. 1. NY; [6] Perry RH, Chlton CH. Chemcal engneers handbook. 5th ed. NY: McGraw-Hll; [7] Krscher O, Kast W. 3rd ed. Dryng technology-the scentfc background of dryng technology, vol. 1. Berln, Hedelberg: Sprnger; [8] Fred C, Adler J. In: Cambel, Fenn, edtors. Transport propertes n gases. Evanston, Ill: Northwestern Unversty Press; p [9] Vargaftk NB. Handbook of physcal propertes of lquds and gasespure substances and mxtures. Washngton: Hemsphere; [30] Kestn J, Whtelaw JH. The vscosty of dry and humd ar. Int J Heat Mass Transfer 1964: [31] Gruss H, Schmck H. Scentfc publcaton of the Semens Company, vol. 7(0);198. [3] Touloukan YS, Lley PE, Saxena SC. Thermal conductvty non metallc lquds and gases, vol. 3. NY: IFI/Plenum Data Corporaton; 1970.